// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2016 Benoit Steiner (benoit.steiner.goog@gmail.com)
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_PACKET_MATH_AVX512_H
#define EIGEN_PACKET_MATH_AVX512_H

namespace Eigen {

namespace internal {

#ifndef EIGEN_CACHEFRIENDLY_PRODUCT_THRESHOLD
#define EIGEN_CACHEFRIENDLY_PRODUCT_THRESHOLD 8
#endif

#ifndef EIGEN_ARCH_DEFAULT_NUMBER_OF_REGISTERS
#define EIGEN_ARCH_DEFAULT_NUMBER_OF_REGISTERS 32
#endif

#ifdef EIGEN_VECTORIZE_FMA
#ifndef EIGEN_HAS_SINGLE_INSTRUCTION_MADD
#define EIGEN_HAS_SINGLE_INSTRUCTION_MADD
#endif
#endif

    typedef __m512 Packet16f;
    typedef __m512i Packet16i;
    typedef __m512d Packet8d;
    typedef eigen_packet_wrapper<__m256i, 1> Packet16h;
    typedef eigen_packet_wrapper<__m256i, 2> Packet16bf;

    template <> struct is_arithmetic<__m512>
    {
        enum
        {
            value = true
        };
    };
    template <> struct is_arithmetic<__m512i>
    {
        enum
        {
            value = true
        };
    };
    template <> struct is_arithmetic<__m512d>
    {
        enum
        {
            value = true
        };
    };

    template <> struct is_arithmetic<Packet16h>
    {
        enum
        {
            value = true
        };
    };

    template <> struct packet_traits<half> : default_packet_traits
    {
        typedef Packet16h type;
        // There is no half-size packet for Packet16h.
        typedef Packet16h half;
        enum
        {
            Vectorizable = 1,
            AlignedOnScalar = 1,
            size = 16,
            HasHalfPacket = 1,

            HasCmp = 1,
            HasAdd = 1,
            HasSub = 1,
            HasMul = 1,
            HasDiv = 1,
            HasNegate = 1,
            HasAbs = 1,
            HasAbs2 = 0,
            HasMin = 1,
            HasMax = 1,
            HasConj = 1,
            HasSetLinear = 0,
            HasLog = 1,
            HasLog1p = 1,
            HasExpm1 = 1,
            HasExp = 1,
            HasSqrt = 1,
            HasRsqrt = 1,
            HasSin = EIGEN_FAST_MATH,
            HasCos = EIGEN_FAST_MATH,
            HasTanh = EIGEN_FAST_MATH,
            HasErf = EIGEN_FAST_MATH,
            HasBlend = 0,
            HasRound = 1,
            HasFloor = 1,
            HasCeil = 1,
            HasRint = 1,
            HasBessel = 1,
            HasNdtri = 1
        };
    };

    template <> struct packet_traits<float> : default_packet_traits
    {
        typedef Packet16f type;
        typedef Packet8f half;
        enum
        {
            Vectorizable = 1,
            AlignedOnScalar = 1,
            size = 16,
            HasHalfPacket = 1,

            HasAbs = 1,
            HasMin = 1,
            HasMax = 1,
            HasConj = 1,
            HasBlend = 0,
            HasSin = EIGEN_FAST_MATH,
            HasCos = EIGEN_FAST_MATH,
#if EIGEN_GNUC_AT_LEAST(5, 3) || (!EIGEN_COMP_GNUC_STRICT)
            HasLog = 1,
            HasLog1p = 1,
            HasExpm1 = 1,
            HasNdtri = 1,
            HasBessel = 1,
            HasExp = 1,
            HasSqrt = EIGEN_FAST_MATH,
            HasRsqrt = EIGEN_FAST_MATH,
            HasTanh = EIGEN_FAST_MATH,
            HasErf = EIGEN_FAST_MATH,
#endif
            HasCmp = 1,
            HasDiv = 1,
            HasRound = 1,
            HasFloor = 1,
            HasCeil = 1,
            HasRint = 1
        };
    };
    template <> struct packet_traits<double> : default_packet_traits
    {
        typedef Packet8d type;
        typedef Packet4d half;
        enum
        {
            Vectorizable = 1,
            AlignedOnScalar = 1,
            size = 8,
            HasHalfPacket = 1,
#if EIGEN_GNUC_AT_LEAST(5, 3) || (!EIGEN_COMP_GNUC_STRICT)
            HasLog = 1,
            HasExp = 1,
            HasSqrt = EIGEN_FAST_MATH,
            HasRsqrt = EIGEN_FAST_MATH,
#endif
            HasCmp = 1,
            HasDiv = 1,
            HasRound = 1,
            HasFloor = 1,
            HasCeil = 1,
            HasRint = 1
        };
    };

    /* TODO Implement AVX512 for integers
template<> struct packet_traits<int>    : default_packet_traits
{
  typedef Packet16i type;
  enum {
    Vectorizable = 1,
    AlignedOnScalar = 1,
    size=8
  };
};
*/

    template <> struct unpacket_traits<Packet16f>
    {
        typedef float type;
        typedef Packet8f half;
        typedef Packet16i integer_packet;
        typedef uint16_t mask_t;
        enum
        {
            size = 16,
            alignment = Aligned64,
            vectorizable = true,
            masked_load_available = true,
            masked_store_available = true
        };
    };
    template <> struct unpacket_traits<Packet8d>
    {
        typedef double type;
        typedef Packet4d half;
        enum
        {
            size = 8,
            alignment = Aligned64,
            vectorizable = true,
            masked_load_available = false,
            masked_store_available = false
        };
    };
    template <> struct unpacket_traits<Packet16i>
    {
        typedef int type;
        typedef Packet8i half;
        enum
        {
            size = 16,
            alignment = Aligned64,
            vectorizable = false,
            masked_load_available = false,
            masked_store_available = false
        };
    };

    template <> struct unpacket_traits<Packet16h>
    {
        typedef Eigen::half type;
        typedef Packet8h half;
        enum
        {
            size = 16,
            alignment = Aligned32,
            vectorizable = true,
            masked_load_available = false,
            masked_store_available = false
        };
    };

    template <> EIGEN_STRONG_INLINE Packet16f pset1<Packet16f>(const float& from) { return _mm512_set1_ps(from); }
    template <> EIGEN_STRONG_INLINE Packet8d pset1<Packet8d>(const double& from) { return _mm512_set1_pd(from); }
    template <> EIGEN_STRONG_INLINE Packet16i pset1<Packet16i>(const int& from) { return _mm512_set1_epi32(from); }

    template <> EIGEN_STRONG_INLINE Packet16f pset1frombits<Packet16f>(unsigned int from) { return _mm512_castsi512_ps(_mm512_set1_epi32(from)); }

    template <> EIGEN_STRONG_INLINE Packet8d pset1frombits<Packet8d>(const numext::uint64_t from) { return _mm512_castsi512_pd(_mm512_set1_epi64(from)); }

    template <> EIGEN_STRONG_INLINE Packet16f pzero(const Packet16f& /*a*/) { return _mm512_setzero_ps(); }
    template <> EIGEN_STRONG_INLINE Packet8d pzero(const Packet8d& /*a*/) { return _mm512_setzero_pd(); }
    template <> EIGEN_STRONG_INLINE Packet16i pzero(const Packet16i& /*a*/) { return _mm512_setzero_si512(); }

    template <> EIGEN_STRONG_INLINE Packet16f peven_mask(const Packet16f& /*a*/)
    {
        return _mm512_castsi512_ps(_mm512_set_epi32(0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1));
    }
    template <> EIGEN_STRONG_INLINE Packet16i peven_mask(const Packet16i& /*a*/)
    {
        return _mm512_set_epi32(0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1);
    }
    template <> EIGEN_STRONG_INLINE Packet8d peven_mask(const Packet8d& /*a*/)
    {
        return _mm512_castsi512_pd(_mm512_set_epi32(0, 0, -1, -1, 0, 0, -1, -1, 0, 0, -1, -1, 0, 0, -1, -1));
    }

    template <> EIGEN_STRONG_INLINE Packet16f pload1<Packet16f>(const float* from) { return _mm512_broadcastss_ps(_mm_load_ps1(from)); }
    template <> EIGEN_STRONG_INLINE Packet8d pload1<Packet8d>(const double* from) { return _mm512_set1_pd(*from); }

    template <> EIGEN_STRONG_INLINE Packet16f plset<Packet16f>(const float& a)
    {
        return _mm512_add_ps(_mm512_set1_ps(a),
                             _mm512_set_ps(15.0f, 14.0f, 13.0f, 12.0f, 11.0f, 10.0f, 9.0f, 8.0f, 7.0f, 6.0f, 5.0f, 4.0f, 3.0f, 2.0f, 1.0f, 0.0f));
    }
    template <> EIGEN_STRONG_INLINE Packet8d plset<Packet8d>(const double& a)
    {
        return _mm512_add_pd(_mm512_set1_pd(a), _mm512_set_pd(7.0, 6.0, 5.0, 4.0, 3.0, 2.0, 1.0, 0.0));
    }

    template <> EIGEN_STRONG_INLINE Packet16f padd<Packet16f>(const Packet16f& a, const Packet16f& b) { return _mm512_add_ps(a, b); }
    template <> EIGEN_STRONG_INLINE Packet8d padd<Packet8d>(const Packet8d& a, const Packet8d& b) { return _mm512_add_pd(a, b); }
    template <> EIGEN_STRONG_INLINE Packet16i padd<Packet16i>(const Packet16i& a, const Packet16i& b) { return _mm512_add_epi32(a, b); }

    template <> EIGEN_STRONG_INLINE Packet16f psub<Packet16f>(const Packet16f& a, const Packet16f& b) { return _mm512_sub_ps(a, b); }
    template <> EIGEN_STRONG_INLINE Packet8d psub<Packet8d>(const Packet8d& a, const Packet8d& b) { return _mm512_sub_pd(a, b); }
    template <> EIGEN_STRONG_INLINE Packet16i psub<Packet16i>(const Packet16i& a, const Packet16i& b) { return _mm512_sub_epi32(a, b); }

    template <> EIGEN_STRONG_INLINE Packet16f pnegate(const Packet16f& a) { return _mm512_sub_ps(_mm512_set1_ps(0.0), a); }
    template <> EIGEN_STRONG_INLINE Packet8d pnegate(const Packet8d& a) { return _mm512_sub_pd(_mm512_set1_pd(0.0), a); }

    template <> EIGEN_STRONG_INLINE Packet16f pconj(const Packet16f& a) { return a; }
    template <> EIGEN_STRONG_INLINE Packet8d pconj(const Packet8d& a) { return a; }
    template <> EIGEN_STRONG_INLINE Packet16i pconj(const Packet16i& a) { return a; }

    template <> EIGEN_STRONG_INLINE Packet16f pmul<Packet16f>(const Packet16f& a, const Packet16f& b) { return _mm512_mul_ps(a, b); }
    template <> EIGEN_STRONG_INLINE Packet8d pmul<Packet8d>(const Packet8d& a, const Packet8d& b) { return _mm512_mul_pd(a, b); }
    template <> EIGEN_STRONG_INLINE Packet16i pmul<Packet16i>(const Packet16i& a, const Packet16i& b) { return _mm512_mullo_epi32(a, b); }

    template <> EIGEN_STRONG_INLINE Packet16f pdiv<Packet16f>(const Packet16f& a, const Packet16f& b) { return _mm512_div_ps(a, b); }
    template <> EIGEN_STRONG_INLINE Packet8d pdiv<Packet8d>(const Packet8d& a, const Packet8d& b) { return _mm512_div_pd(a, b); }

#ifdef EIGEN_VECTORIZE_FMA
    template <> EIGEN_STRONG_INLINE Packet16f pmadd(const Packet16f& a, const Packet16f& b, const Packet16f& c) { return _mm512_fmadd_ps(a, b, c); }
    template <> EIGEN_STRONG_INLINE Packet8d pmadd(const Packet8d& a, const Packet8d& b, const Packet8d& c) { return _mm512_fmadd_pd(a, b, c); }
#endif

    template <> EIGEN_DEVICE_FUNC inline Packet16f pselect(const Packet16f& mask, const Packet16f& a, const Packet16f& b)
    {
        __mmask16 mask16 = _mm512_cmp_epi32_mask(_mm512_castps_si512(mask), _mm512_setzero_epi32(), _MM_CMPINT_EQ);
        return _mm512_mask_blend_ps(mask16, a, b);
    }

    template <> EIGEN_DEVICE_FUNC inline Packet8d pselect(const Packet8d& mask, const Packet8d& a, const Packet8d& b)
    {
        __mmask8 mask8 = _mm512_cmp_epi64_mask(_mm512_castpd_si512(mask), _mm512_setzero_epi32(), _MM_CMPINT_EQ);
        return _mm512_mask_blend_pd(mask8, a, b);
    }

    template <> EIGEN_STRONG_INLINE Packet16f pmin<Packet16f>(const Packet16f& a, const Packet16f& b)
    {
        // Arguments are reversed to match NaN propagation behavior of std::min.
        return _mm512_min_ps(b, a);
    }
    template <> EIGEN_STRONG_INLINE Packet8d pmin<Packet8d>(const Packet8d& a, const Packet8d& b)
    {
        // Arguments are reversed to match NaN propagation behavior of std::min.
        return _mm512_min_pd(b, a);
    }

    template <> EIGEN_STRONG_INLINE Packet16f pmax<Packet16f>(const Packet16f& a, const Packet16f& b)
    {
        // Arguments are reversed to match NaN propagation behavior of std::max.
        return _mm512_max_ps(b, a);
    }
    template <> EIGEN_STRONG_INLINE Packet8d pmax<Packet8d>(const Packet8d& a, const Packet8d& b)
    {
        // Arguments are reversed to match NaN propagation behavior of std::max.
        return _mm512_max_pd(b, a);
    }

    // Add specializations for min/max with prescribed NaN progation.
    template <> EIGEN_STRONG_INLINE Packet16f pmin<PropagateNumbers, Packet16f>(const Packet16f& a, const Packet16f& b)
    {
        return pminmax_propagate_numbers(a, b, pmin<Packet16f>);
    }
    template <> EIGEN_STRONG_INLINE Packet8d pmin<PropagateNumbers, Packet8d>(const Packet8d& a, const Packet8d& b)
    {
        return pminmax_propagate_numbers(a, b, pmin<Packet8d>);
    }
    template <> EIGEN_STRONG_INLINE Packet16f pmax<PropagateNumbers, Packet16f>(const Packet16f& a, const Packet16f& b)
    {
        return pminmax_propagate_numbers(a, b, pmax<Packet16f>);
    }
    template <> EIGEN_STRONG_INLINE Packet8d pmax<PropagateNumbers, Packet8d>(const Packet8d& a, const Packet8d& b)
    {
        return pminmax_propagate_numbers(a, b, pmax<Packet8d>);
    }
    template <> EIGEN_STRONG_INLINE Packet16f pmin<PropagateNaN, Packet16f>(const Packet16f& a, const Packet16f& b)
    {
        return pminmax_propagate_nan(a, b, pmin<Packet16f>);
    }
    template <> EIGEN_STRONG_INLINE Packet8d pmin<PropagateNaN, Packet8d>(const Packet8d& a, const Packet8d& b)
    {
        return pminmax_propagate_nan(a, b, pmin<Packet8d>);
    }
    template <> EIGEN_STRONG_INLINE Packet16f pmax<PropagateNaN, Packet16f>(const Packet16f& a, const Packet16f& b)
    {
        return pminmax_propagate_nan(a, b, pmax<Packet16f>);
    }
    template <> EIGEN_STRONG_INLINE Packet8d pmax<PropagateNaN, Packet8d>(const Packet8d& a, const Packet8d& b)
    {
        return pminmax_propagate_nan(a, b, pmax<Packet8d>);
    }

#ifdef EIGEN_VECTORIZE_AVX512DQ
    template <int I_> EIGEN_STRONG_INLINE Packet8f extract256(Packet16f x) { return _mm512_extractf32x8_ps(x, I_); }
    template <int I_> EIGEN_STRONG_INLINE Packet2d extract128(Packet8d x) { return _mm512_extractf64x2_pd(x, I_); }
    EIGEN_STRONG_INLINE Packet16f cat256(Packet8f a, Packet8f b) { return _mm512_insertf32x8(_mm512_castps256_ps512(a), b, 1); }
#else
    // AVX512F does not define _mm512_extractf32x8_ps to extract _m256 from _m512
    template <int I_> EIGEN_STRONG_INLINE Packet8f extract256(Packet16f x)
    {
        return _mm256_castsi256_ps(_mm512_extracti64x4_epi64(_mm512_castps_si512(x), I_));
    }

    // AVX512F does not define _mm512_extractf64x2_pd to extract _m128 from _m512
    template <int I_> EIGEN_STRONG_INLINE Packet2d extract128(Packet8d x) { return _mm_castsi128_pd(_mm512_extracti32x4_epi32(_mm512_castpd_si512(x), I_)); }

    EIGEN_STRONG_INLINE Packet16f cat256(Packet8f a, Packet8f b)
    {
        return _mm512_castsi512_ps(_mm512_inserti64x4(_mm512_castsi256_si512(_mm256_castps_si256(a)), _mm256_castps_si256(b), 1));
    }
#endif

    // Helper function for bit packing snippet of low precision comparison.
    // It packs the flags from 32x16 to 16x16.
    EIGEN_STRONG_INLINE __m256i Pack32To16(Packet16f rf)
    {
        // Split data into small pieces and handle with AVX instructions
        // to guarantee internal order of vector.
        // Operation:
        //   dst[15:0]    := Saturate16(rf[31:0])
        //   dst[31:16]   := Saturate16(rf[63:32])
        //   ...
        //   dst[255:240] := Saturate16(rf[255:224])
        __m256i lo = _mm256_castps_si256(extract256<0>(rf));
        __m256i hi = _mm256_castps_si256(extract256<1>(rf));
        __m128i result_lo = _mm_packs_epi32(_mm256_extractf128_si256(lo, 0), _mm256_extractf128_si256(lo, 1));
        __m128i result_hi = _mm_packs_epi32(_mm256_extractf128_si256(hi, 0), _mm256_extractf128_si256(hi, 1));
        return _mm256_insertf128_si256(_mm256_castsi128_si256(result_lo), result_hi, 1);
    }

    template <> EIGEN_STRONG_INLINE Packet16f pcmp_eq(const Packet16f& a, const Packet16f& b)
    {
        __mmask16 mask = _mm512_cmp_ps_mask(a, b, _CMP_EQ_OQ);
        return _mm512_castsi512_ps(_mm512_mask_set1_epi32(_mm512_set1_epi32(0), mask, 0xffffffffu));
    }
    template <> EIGEN_STRONG_INLINE Packet16f pcmp_le(const Packet16f& a, const Packet16f& b)
    {
        __mmask16 mask = _mm512_cmp_ps_mask(a, b, _CMP_LE_OQ);
        return _mm512_castsi512_ps(_mm512_mask_set1_epi32(_mm512_set1_epi32(0), mask, 0xffffffffu));
    }

    template <> EIGEN_STRONG_INLINE Packet16f pcmp_lt(const Packet16f& a, const Packet16f& b)
    {
        __mmask16 mask = _mm512_cmp_ps_mask(a, b, _CMP_LT_OQ);
        return _mm512_castsi512_ps(_mm512_mask_set1_epi32(_mm512_set1_epi32(0), mask, 0xffffffffu));
    }

    template <> EIGEN_STRONG_INLINE Packet16f pcmp_lt_or_nan(const Packet16f& a, const Packet16f& b)
    {
        __mmask16 mask = _mm512_cmp_ps_mask(a, b, _CMP_NGE_UQ);
        return _mm512_castsi512_ps(_mm512_mask_set1_epi32(_mm512_set1_epi32(0), mask, 0xffffffffu));
    }

    template <> EIGEN_STRONG_INLINE Packet16i pcmp_eq(const Packet16i& a, const Packet16i& b)
    {
        __mmask16 mask = _mm512_cmp_epi32_mask(a, b, _CMP_EQ_OQ);
        return _mm512_mask_set1_epi32(_mm512_set1_epi32(0), mask, 0xffffffffu);
    }

    template <> EIGEN_STRONG_INLINE Packet8d pcmp_eq(const Packet8d& a, const Packet8d& b)
    {
        __mmask8 mask = _mm512_cmp_pd_mask(a, b, _CMP_EQ_OQ);
        return _mm512_castsi512_pd(_mm512_mask_set1_epi64(_mm512_set1_epi64(0), mask, 0xffffffffffffffffu));
    }
    template <> EIGEN_STRONG_INLINE Packet8d pcmp_le(const Packet8d& a, const Packet8d& b)
    {
        __mmask8 mask = _mm512_cmp_pd_mask(a, b, _CMP_LE_OQ);
        return _mm512_castsi512_pd(_mm512_mask_set1_epi64(_mm512_set1_epi64(0), mask, 0xffffffffffffffffu));
    }
    template <> EIGEN_STRONG_INLINE Packet8d pcmp_lt(const Packet8d& a, const Packet8d& b)
    {
        __mmask8 mask = _mm512_cmp_pd_mask(a, b, _CMP_LT_OQ);
        return _mm512_castsi512_pd(_mm512_mask_set1_epi64(_mm512_set1_epi64(0), mask, 0xffffffffffffffffu));
    }
    template <> EIGEN_STRONG_INLINE Packet8d pcmp_lt_or_nan(const Packet8d& a, const Packet8d& b)
    {
        __mmask8 mask = _mm512_cmp_pd_mask(a, b, _CMP_NGE_UQ);
        return _mm512_castsi512_pd(_mm512_mask_set1_epi64(_mm512_set1_epi64(0), mask, 0xffffffffffffffffu));
    }

    template <> EIGEN_STRONG_INLINE Packet16f print<Packet16f>(const Packet16f& a) { return _mm512_roundscale_ps(a, _MM_FROUND_CUR_DIRECTION); }
    template <> EIGEN_STRONG_INLINE Packet8d print<Packet8d>(const Packet8d& a) { return _mm512_roundscale_pd(a, _MM_FROUND_CUR_DIRECTION); }

    template <> EIGEN_STRONG_INLINE Packet16f pceil<Packet16f>(const Packet16f& a) { return _mm512_roundscale_ps(a, _MM_FROUND_TO_POS_INF); }
    template <> EIGEN_STRONG_INLINE Packet8d pceil<Packet8d>(const Packet8d& a) { return _mm512_roundscale_pd(a, _MM_FROUND_TO_POS_INF); }

    template <> EIGEN_STRONG_INLINE Packet16f pfloor<Packet16f>(const Packet16f& a) { return _mm512_roundscale_ps(a, _MM_FROUND_TO_NEG_INF); }
    template <> EIGEN_STRONG_INLINE Packet8d pfloor<Packet8d>(const Packet8d& a) { return _mm512_roundscale_pd(a, _MM_FROUND_TO_NEG_INF); }

    template <> EIGEN_STRONG_INLINE Packet16i ptrue<Packet16i>(const Packet16i& /*a*/) { return _mm512_set1_epi32(0xffffffffu); }

    template <> EIGEN_STRONG_INLINE Packet16f ptrue<Packet16f>(const Packet16f& a) { return _mm512_castsi512_ps(ptrue<Packet16i>(_mm512_castps_si512(a))); }

    template <> EIGEN_STRONG_INLINE Packet8d ptrue<Packet8d>(const Packet8d& a) { return _mm512_castsi512_pd(ptrue<Packet16i>(_mm512_castpd_si512(a))); }

    template <> EIGEN_STRONG_INLINE Packet16i pand<Packet16i>(const Packet16i& a, const Packet16i& b) { return _mm512_and_si512(a, b); }

    template <> EIGEN_STRONG_INLINE Packet16f pand<Packet16f>(const Packet16f& a, const Packet16f& b)
    {
#ifdef EIGEN_VECTORIZE_AVX512DQ
        return _mm512_and_ps(a, b);
#else
        return _mm512_castsi512_ps(pand(_mm512_castps_si512(a), _mm512_castps_si512(b)));
#endif
    }
    template <> EIGEN_STRONG_INLINE Packet8d pand<Packet8d>(const Packet8d& a, const Packet8d& b)
    {
#ifdef EIGEN_VECTORIZE_AVX512DQ
        return _mm512_and_pd(a, b);
#else
        Packet8d res = _mm512_undefined_pd();
        Packet4d lane0_a = _mm512_extractf64x4_pd(a, 0);
        Packet4d lane0_b = _mm512_extractf64x4_pd(b, 0);
        res = _mm512_insertf64x4(res, _mm256_and_pd(lane0_a, lane0_b), 0);

        Packet4d lane1_a = _mm512_extractf64x4_pd(a, 1);
        Packet4d lane1_b = _mm512_extractf64x4_pd(b, 1);
        return _mm512_insertf64x4(res, _mm256_and_pd(lane1_a, lane1_b), 1);
#endif
    }

    template <> EIGEN_STRONG_INLINE Packet16i por<Packet16i>(const Packet16i& a, const Packet16i& b) { return _mm512_or_si512(a, b); }

    template <> EIGEN_STRONG_INLINE Packet16f por<Packet16f>(const Packet16f& a, const Packet16f& b)
    {
#ifdef EIGEN_VECTORIZE_AVX512DQ
        return _mm512_or_ps(a, b);
#else
        return _mm512_castsi512_ps(por(_mm512_castps_si512(a), _mm512_castps_si512(b)));
#endif
    }

    template <> EIGEN_STRONG_INLINE Packet8d por<Packet8d>(const Packet8d& a, const Packet8d& b)
    {
#ifdef EIGEN_VECTORIZE_AVX512DQ
        return _mm512_or_pd(a, b);
#else
        return _mm512_castsi512_pd(por(_mm512_castpd_si512(a), _mm512_castpd_si512(b)));
#endif
    }

    template <> EIGEN_STRONG_INLINE Packet16i pxor<Packet16i>(const Packet16i& a, const Packet16i& b) { return _mm512_xor_si512(a, b); }

    template <> EIGEN_STRONG_INLINE Packet16f pxor<Packet16f>(const Packet16f& a, const Packet16f& b)
    {
#ifdef EIGEN_VECTORIZE_AVX512DQ
        return _mm512_xor_ps(a, b);
#else
        return _mm512_castsi512_ps(pxor(_mm512_castps_si512(a), _mm512_castps_si512(b)));
#endif
    }

    template <> EIGEN_STRONG_INLINE Packet8d pxor<Packet8d>(const Packet8d& a, const Packet8d& b)
    {
#ifdef EIGEN_VECTORIZE_AVX512DQ
        return _mm512_xor_pd(a, b);
#else
        return _mm512_castsi512_pd(pxor(_mm512_castpd_si512(a), _mm512_castpd_si512(b)));
#endif
    }

    template <> EIGEN_STRONG_INLINE Packet16i pandnot<Packet16i>(const Packet16i& a, const Packet16i& b) { return _mm512_andnot_si512(b, a); }

    template <> EIGEN_STRONG_INLINE Packet16f pandnot<Packet16f>(const Packet16f& a, const Packet16f& b)
    {
#ifdef EIGEN_VECTORIZE_AVX512DQ
        return _mm512_andnot_ps(b, a);
#else
        return _mm512_castsi512_ps(pandnot(_mm512_castps_si512(a), _mm512_castps_si512(b)));
#endif
    }
    template <> EIGEN_STRONG_INLINE Packet8d pandnot<Packet8d>(const Packet8d& a, const Packet8d& b)
    {
#ifdef EIGEN_VECTORIZE_AVX512DQ
        return _mm512_andnot_pd(b, a);
#else
        return _mm512_castsi512_pd(pandnot(_mm512_castpd_si512(a), _mm512_castpd_si512(b)));
#endif
    }

    template <> EIGEN_STRONG_INLINE Packet16f pround<Packet16f>(const Packet16f& a)
    {
        // Work-around for default std::round rounding mode.
        const Packet16f mask = pset1frombits<Packet16f>(static_cast<numext::uint32_t>(0x80000000u));
        const Packet16f prev0dot5 = pset1frombits<Packet16f>(static_cast<numext::uint32_t>(0x3EFFFFFFu));
        return _mm512_roundscale_ps(padd(por(pand(a, mask), prev0dot5), a), _MM_FROUND_TO_ZERO);
    }
    template <> EIGEN_STRONG_INLINE Packet8d pround<Packet8d>(const Packet8d& a)
    {
        // Work-around for default std::round rounding mode.
        const Packet8d mask = pset1frombits<Packet8d>(static_cast<numext::uint64_t>(0x8000000000000000ull));
        const Packet8d prev0dot5 = pset1frombits<Packet8d>(static_cast<numext::uint64_t>(0x3FDFFFFFFFFFFFFFull));
        return _mm512_roundscale_pd(padd(por(pand(a, mask), prev0dot5), a), _MM_FROUND_TO_ZERO);
    }

    template <int N> EIGEN_STRONG_INLINE Packet16i parithmetic_shift_right(Packet16i a) { return _mm512_srai_epi32(a, N); }

    template <int N> EIGEN_STRONG_INLINE Packet16i plogical_shift_right(Packet16i a) { return _mm512_srli_epi32(a, N); }

    template <int N> EIGEN_STRONG_INLINE Packet16i plogical_shift_left(Packet16i a) { return _mm512_slli_epi32(a, N); }

    template <> EIGEN_STRONG_INLINE Packet16f pload<Packet16f>(const float* from) { EIGEN_DEBUG_ALIGNED_LOAD return _mm512_load_ps(from); }
    template <> EIGEN_STRONG_INLINE Packet8d pload<Packet8d>(const double* from) { EIGEN_DEBUG_ALIGNED_LOAD return _mm512_load_pd(from); }
    template <> EIGEN_STRONG_INLINE Packet16i pload<Packet16i>(const int* from)
    {
        EIGEN_DEBUG_ALIGNED_LOAD return _mm512_load_si512(reinterpret_cast<const __m512i*>(from));
    }

    template <> EIGEN_STRONG_INLINE Packet16f ploadu<Packet16f>(const float* from) { EIGEN_DEBUG_UNALIGNED_LOAD return _mm512_loadu_ps(from); }
    template <> EIGEN_STRONG_INLINE Packet8d ploadu<Packet8d>(const double* from) { EIGEN_DEBUG_UNALIGNED_LOAD return _mm512_loadu_pd(from); }
    template <> EIGEN_STRONG_INLINE Packet16i ploadu<Packet16i>(const int* from)
    {
        EIGEN_DEBUG_UNALIGNED_LOAD return _mm512_loadu_si512(reinterpret_cast<const __m512i*>(from));
    }

    template <> EIGEN_STRONG_INLINE Packet16f ploadu<Packet16f>(const float* from, uint16_t umask)
    {
        __mmask16 mask = static_cast<__mmask16>(umask);
        EIGEN_DEBUG_UNALIGNED_LOAD return _mm512_maskz_loadu_ps(mask, from);
    }

    // Loads 8 floats from memory a returns the packet
    // {a0, a0  a1, a1, a2, a2, a3, a3, a4, a4, a5, a5, a6, a6, a7, a7}
    template <> EIGEN_STRONG_INLINE Packet16f ploaddup<Packet16f>(const float* from)
    {
        // an unaligned load is required here as there is no requirement
        // on the alignment of input pointer 'from'
        __m256i low_half = _mm256_loadu_si256(reinterpret_cast<const __m256i*>(from));
        __m512 even_elements = _mm512_castsi512_ps(_mm512_cvtepu32_epi64(low_half));
        __m512 pairs = _mm512_permute_ps(even_elements, _MM_SHUFFLE(2, 2, 0, 0));
        return pairs;
    }

#ifdef EIGEN_VECTORIZE_AVX512DQ
    // FIXME: this does not look optimal, better load a Packet4d and shuffle...
    // Loads 4 doubles from memory a returns the packet {a0, a0  a1, a1, a2, a2, a3,
    // a3}
    template <> EIGEN_STRONG_INLINE Packet8d ploaddup<Packet8d>(const double* from)
    {
        __m512d x = _mm512_setzero_pd();
        x = _mm512_insertf64x2(x, _mm_loaddup_pd(&from[0]), 0);
        x = _mm512_insertf64x2(x, _mm_loaddup_pd(&from[1]), 1);
        x = _mm512_insertf64x2(x, _mm_loaddup_pd(&from[2]), 2);
        x = _mm512_insertf64x2(x, _mm_loaddup_pd(&from[3]), 3);
        return x;
    }
#else
    template <> EIGEN_STRONG_INLINE Packet8d ploaddup<Packet8d>(const double* from)
    {
        __m512d x = _mm512_setzero_pd();
        x = _mm512_mask_broadcastsd_pd(x, 0x3 << 0, _mm_load_sd(from + 0));
        x = _mm512_mask_broadcastsd_pd(x, 0x3 << 2, _mm_load_sd(from + 1));
        x = _mm512_mask_broadcastsd_pd(x, 0x3 << 4, _mm_load_sd(from + 2));
        x = _mm512_mask_broadcastsd_pd(x, 0x3 << 6, _mm_load_sd(from + 3));
        return x;
    }
#endif

    // Loads 4 floats from memory a returns the packet
    // {a0, a0  a0, a0, a1, a1, a1, a1, a2, a2, a2, a2, a3, a3, a3, a3}
    template <> EIGEN_STRONG_INLINE Packet16f ploadquad<Packet16f>(const float* from)
    {
        Packet16f tmp = _mm512_castps128_ps512(ploadu<Packet4f>(from));
        const Packet16i scatter_mask = _mm512_set_epi32(3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0);
        return _mm512_permutexvar_ps(scatter_mask, tmp);
    }

    // Loads 2 doubles from memory a returns the packet
    // {a0, a0  a0, a0, a1, a1, a1, a1}
    template <> EIGEN_STRONG_INLINE Packet8d ploadquad<Packet8d>(const double* from)
    {
        __m256d lane0 = _mm256_set1_pd(*from);
        __m256d lane1 = _mm256_set1_pd(*(from + 1));
        __m512d tmp = _mm512_undefined_pd();
        tmp = _mm512_insertf64x4(tmp, lane0, 0);
        return _mm512_insertf64x4(tmp, lane1, 1);
    }

    template <> EIGEN_STRONG_INLINE void pstore<float>(float* to, const Packet16f& from) { EIGEN_DEBUG_ALIGNED_STORE _mm512_store_ps(to, from); }
    template <> EIGEN_STRONG_INLINE void pstore<double>(double* to, const Packet8d& from) { EIGEN_DEBUG_ALIGNED_STORE _mm512_store_pd(to, from); }
    template <> EIGEN_STRONG_INLINE void pstore<int>(int* to, const Packet16i& from)
    {
        EIGEN_DEBUG_ALIGNED_STORE _mm512_storeu_si512(reinterpret_cast<__m512i*>(to), from);
    }

    template <> EIGEN_STRONG_INLINE void pstoreu<float>(float* to, const Packet16f& from) { EIGEN_DEBUG_UNALIGNED_STORE _mm512_storeu_ps(to, from); }
    template <> EIGEN_STRONG_INLINE void pstoreu<double>(double* to, const Packet8d& from) { EIGEN_DEBUG_UNALIGNED_STORE _mm512_storeu_pd(to, from); }
    template <> EIGEN_STRONG_INLINE void pstoreu<int>(int* to, const Packet16i& from)
    {
        EIGEN_DEBUG_UNALIGNED_STORE _mm512_storeu_si512(reinterpret_cast<__m512i*>(to), from);
    }
    template <> EIGEN_STRONG_INLINE void pstoreu<float>(float* to, const Packet16f& from, uint16_t umask)
    {
        __mmask16 mask = static_cast<__mmask16>(umask);
        EIGEN_DEBUG_UNALIGNED_STORE return _mm512_mask_storeu_ps(to, mask, from);
    }

    template <> EIGEN_DEVICE_FUNC inline Packet16f pgather<float, Packet16f>(const float* from, Index stride)
    {
        Packet16i stride_vector = _mm512_set1_epi32(convert_index<int>(stride));
        Packet16i stride_multiplier = _mm512_set_epi32(15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0);
        Packet16i indices = _mm512_mullo_epi32(stride_vector, stride_multiplier);

        return _mm512_i32gather_ps(indices, from, 4);
    }
    template <> EIGEN_DEVICE_FUNC inline Packet8d pgather<double, Packet8d>(const double* from, Index stride)
    {
        Packet8i stride_vector = _mm256_set1_epi32(convert_index<int>(stride));
        Packet8i stride_multiplier = _mm256_set_epi32(7, 6, 5, 4, 3, 2, 1, 0);
        Packet8i indices = _mm256_mullo_epi32(stride_vector, stride_multiplier);

        return _mm512_i32gather_pd(indices, from, 8);
    }

    template <> EIGEN_DEVICE_FUNC inline void pscatter<float, Packet16f>(float* to, const Packet16f& from, Index stride)
    {
        Packet16i stride_vector = _mm512_set1_epi32(convert_index<int>(stride));
        Packet16i stride_multiplier = _mm512_set_epi32(15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0);
        Packet16i indices = _mm512_mullo_epi32(stride_vector, stride_multiplier);
        _mm512_i32scatter_ps(to, indices, from, 4);
    }
    template <> EIGEN_DEVICE_FUNC inline void pscatter<double, Packet8d>(double* to, const Packet8d& from, Index stride)
    {
        Packet8i stride_vector = _mm256_set1_epi32(convert_index<int>(stride));
        Packet8i stride_multiplier = _mm256_set_epi32(7, 6, 5, 4, 3, 2, 1, 0);
        Packet8i indices = _mm256_mullo_epi32(stride_vector, stride_multiplier);
        _mm512_i32scatter_pd(to, indices, from, 8);
    }

    template <> EIGEN_STRONG_INLINE void pstore1<Packet16f>(float* to, const float& a)
    {
        Packet16f pa = pset1<Packet16f>(a);
        pstore(to, pa);
    }
    template <> EIGEN_STRONG_INLINE void pstore1<Packet8d>(double* to, const double& a)
    {
        Packet8d pa = pset1<Packet8d>(a);
        pstore(to, pa);
    }
    template <> EIGEN_STRONG_INLINE void pstore1<Packet16i>(int* to, const int& a)
    {
        Packet16i pa = pset1<Packet16i>(a);
        pstore(to, pa);
    }

    template <> EIGEN_STRONG_INLINE void prefetch<float>(const float* addr) { _mm_prefetch((SsePrefetchPtrType)(addr), _MM_HINT_T0); }
    template <> EIGEN_STRONG_INLINE void prefetch<double>(const double* addr) { _mm_prefetch((SsePrefetchPtrType)(addr), _MM_HINT_T0); }
    template <> EIGEN_STRONG_INLINE void prefetch<int>(const int* addr) { _mm_prefetch((SsePrefetchPtrType)(addr), _MM_HINT_T0); }

    template <> EIGEN_STRONG_INLINE float pfirst<Packet16f>(const Packet16f& a) { return _mm_cvtss_f32(_mm512_extractf32x4_ps(a, 0)); }
    template <> EIGEN_STRONG_INLINE double pfirst<Packet8d>(const Packet8d& a) { return _mm_cvtsd_f64(_mm256_extractf128_pd(_mm512_extractf64x4_pd(a, 0), 0)); }
    template <> EIGEN_STRONG_INLINE int pfirst<Packet16i>(const Packet16i& a) { return _mm_extract_epi32(_mm512_extracti32x4_epi32(a, 0), 0); }

    template <> EIGEN_STRONG_INLINE Packet16f preverse(const Packet16f& a)
    {
        return _mm512_permutexvar_ps(_mm512_set_epi32(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15), a);
    }

    template <> EIGEN_STRONG_INLINE Packet8d preverse(const Packet8d& a)
    {
        return _mm512_permutexvar_pd(_mm512_set_epi32(0, 0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7), a);
    }

    template <> EIGEN_STRONG_INLINE Packet16f pabs(const Packet16f& a)
    {
        // _mm512_abs_ps intrinsic not found, so hack around it
        return _mm512_castsi512_ps(_mm512_and_si512(_mm512_castps_si512(a), _mm512_set1_epi32(0x7fffffff)));
    }
    template <> EIGEN_STRONG_INLINE Packet8d pabs(const Packet8d& a)
    {
        // _mm512_abs_ps intrinsic not found, so hack around it
        return _mm512_castsi512_pd(_mm512_and_si512(_mm512_castpd_si512(a), _mm512_set1_epi64(0x7fffffffffffffff)));
    }

    template <> EIGEN_STRONG_INLINE Packet16f pfrexp<Packet16f>(const Packet16f& a, Packet16f& exponent) { return pfrexp_generic(a, exponent); }

    // Extract exponent without existence of Packet8l.
    template <> EIGEN_STRONG_INLINE Packet8d pfrexp_generic_get_biased_exponent(const Packet8d& a)
    {
        const Packet8d cst_exp_mask = pset1frombits<Packet8d>(static_cast<uint64_t>(0x7ff0000000000000ull));
#ifdef EIGEN_VECTORIZE_AVX512DQ
        return _mm512_cvtepi64_pd(_mm512_srli_epi64(_mm512_castpd_si512(pand(a, cst_exp_mask)), 52));
#else
        return _mm512_cvtepi32_pd(_mm512_cvtepi64_epi32(_mm512_srli_epi64(_mm512_castpd_si512(pand(a, cst_exp_mask)), 52)));
#endif
    }

    template <> EIGEN_STRONG_INLINE Packet8d pfrexp<Packet8d>(const Packet8d& a, Packet8d& exponent) { return pfrexp_generic(a, exponent); }

    template <> EIGEN_STRONG_INLINE Packet16f pldexp<Packet16f>(const Packet16f& a, const Packet16f& exponent) { return pldexp_generic(a, exponent); }

    template <> EIGEN_STRONG_INLINE Packet8d pldexp<Packet8d>(const Packet8d& a, const Packet8d& exponent)
    {
        // Clamp exponent to [-2099, 2099]
        const Packet8d max_exponent = pset1<Packet8d>(2099.0);
        const Packet8i e = _mm512_cvtpd_epi32(pmin(pmax(exponent, pnegate(max_exponent)), max_exponent));

        // Split 2^e into four factors and multiply.
        const Packet8i bias = pset1<Packet8i>(1023);
        Packet8i b = parithmetic_shift_right<2>(e);  // floor(e/4)

        // 2^b
        const Packet8i permute_idx = _mm256_setr_epi32(0, 4, 1, 5, 2, 6, 3, 7);
        Packet8i hi = _mm256_permutevar8x32_epi32(padd(b, bias), permute_idx);
        Packet8i lo = _mm256_slli_epi64(hi, 52);
        hi = _mm256_slli_epi64(_mm256_srli_epi64(hi, 32), 52);
        Packet8d c = _mm512_castsi512_pd(_mm512_inserti64x4(_mm512_castsi256_si512(lo), hi, 1));
        Packet8d out = pmul(pmul(pmul(a, c), c), c);  // a * 2^(3b)

        // 2^(e - 3b)
        b = psub(psub(psub(e, b), b), b);  // e - 3b
        hi = _mm256_permutevar8x32_epi32(padd(b, bias), permute_idx);
        lo = _mm256_slli_epi64(hi, 52);
        hi = _mm256_slli_epi64(_mm256_srli_epi64(hi, 32), 52);
        c = _mm512_castsi512_pd(_mm512_inserti64x4(_mm512_castsi256_si512(lo), hi, 1));
        out = pmul(out, c);  // a * 2^e
        return out;
    }

#ifdef EIGEN_VECTORIZE_AVX512DQ
// AVX512F does not define _mm512_extractf32x8_ps to extract _m256 from _m512
#define EIGEN_EXTRACT_8f_FROM_16f(INPUT, OUTPUT)          \
    __m256 OUTPUT##_0 = _mm512_extractf32x8_ps(INPUT, 0); \
    __m256 OUTPUT##_1 = _mm512_extractf32x8_ps(INPUT, 1)
#else
#define EIGEN_EXTRACT_8f_FROM_16f(INPUT, OUTPUT)                                                                                             \
    __m256 OUTPUT##_0 = _mm256_insertf128_ps(_mm256_castps128_ps256(_mm512_extractf32x4_ps(INPUT, 0)), _mm512_extractf32x4_ps(INPUT, 1), 1); \
    __m256 OUTPUT##_1 = _mm256_insertf128_ps(_mm256_castps128_ps256(_mm512_extractf32x4_ps(INPUT, 2)), _mm512_extractf32x4_ps(INPUT, 3), 1);
#endif

#ifdef EIGEN_VECTORIZE_AVX512DQ
#define EIGEN_INSERT_8f_INTO_16f(OUTPUT, INPUTA, INPUTB) OUTPUT = _mm512_insertf32x8(_mm512_castps256_ps512(INPUTA), INPUTB, 1);
#else
#define EIGEN_INSERT_8f_INTO_16f(OUTPUT, INPUTA, INPUTB)                      \
    OUTPUT = _mm512_undefined_ps();                                           \
    OUTPUT = _mm512_insertf32x4(OUTPUT, _mm256_extractf128_ps(INPUTA, 0), 0); \
    OUTPUT = _mm512_insertf32x4(OUTPUT, _mm256_extractf128_ps(INPUTA, 1), 1); \
    OUTPUT = _mm512_insertf32x4(OUTPUT, _mm256_extractf128_ps(INPUTB, 0), 2); \
    OUTPUT = _mm512_insertf32x4(OUTPUT, _mm256_extractf128_ps(INPUTB, 1), 3);
#endif

    template <> EIGEN_STRONG_INLINE float predux<Packet16f>(const Packet16f& a)
    {
#ifdef EIGEN_VECTORIZE_AVX512DQ
        __m256 lane0 = _mm512_extractf32x8_ps(a, 0);
        __m256 lane1 = _mm512_extractf32x8_ps(a, 1);
        Packet8f x = _mm256_add_ps(lane0, lane1);
        return predux<Packet8f>(x);
#else
        __m128 lane0 = _mm512_extractf32x4_ps(a, 0);
        __m128 lane1 = _mm512_extractf32x4_ps(a, 1);
        __m128 lane2 = _mm512_extractf32x4_ps(a, 2);
        __m128 lane3 = _mm512_extractf32x4_ps(a, 3);
        __m128 sum = _mm_add_ps(_mm_add_ps(lane0, lane1), _mm_add_ps(lane2, lane3));
        sum = _mm_hadd_ps(sum, sum);
        sum = _mm_hadd_ps(sum, _mm_permute_ps(sum, 1));
        return _mm_cvtss_f32(sum);
#endif
    }
    template <> EIGEN_STRONG_INLINE double predux<Packet8d>(const Packet8d& a)
    {
        __m256d lane0 = _mm512_extractf64x4_pd(a, 0);
        __m256d lane1 = _mm512_extractf64x4_pd(a, 1);
        __m256d sum = _mm256_add_pd(lane0, lane1);
        __m256d tmp0 = _mm256_hadd_pd(sum, _mm256_permute2f128_pd(sum, sum, 1));
        return _mm_cvtsd_f64(_mm256_castpd256_pd128(_mm256_hadd_pd(tmp0, tmp0)));
    }

    template <> EIGEN_STRONG_INLINE Packet8f predux_half_dowto4<Packet16f>(const Packet16f& a)
    {
#ifdef EIGEN_VECTORIZE_AVX512DQ
        __m256 lane0 = _mm512_extractf32x8_ps(a, 0);
        __m256 lane1 = _mm512_extractf32x8_ps(a, 1);
        return _mm256_add_ps(lane0, lane1);
#else
        __m128 lane0 = _mm512_extractf32x4_ps(a, 0);
        __m128 lane1 = _mm512_extractf32x4_ps(a, 1);
        __m128 lane2 = _mm512_extractf32x4_ps(a, 2);
        __m128 lane3 = _mm512_extractf32x4_ps(a, 3);
        __m128 sum0 = _mm_add_ps(lane0, lane2);
        __m128 sum1 = _mm_add_ps(lane1, lane3);
        return _mm256_insertf128_ps(_mm256_castps128_ps256(sum0), sum1, 1);
#endif
    }
    template <> EIGEN_STRONG_INLINE Packet4d predux_half_dowto4<Packet8d>(const Packet8d& a)
    {
        __m256d lane0 = _mm512_extractf64x4_pd(a, 0);
        __m256d lane1 = _mm512_extractf64x4_pd(a, 1);
        return _mm256_add_pd(lane0, lane1);
    }

    template <> EIGEN_STRONG_INLINE float predux_mul<Packet16f>(const Packet16f& a)
    {
//#ifdef EIGEN_VECTORIZE_AVX512DQ
#if 0
  Packet8f lane0 = _mm512_extractf32x8_ps(a, 0);
  Packet8f lane1 = _mm512_extractf32x8_ps(a, 1);
  Packet8f res = pmul(lane0, lane1);
  res = pmul(res, _mm256_permute2f128_ps(res, res, 1));
  res = pmul(res, _mm_permute_ps(res, _MM_SHUFFLE(0, 0, 3, 2)));
  return pfirst(pmul(res, _mm_permute_ps(res, _MM_SHUFFLE(0, 0, 0, 1))));
#else
        __m128 lane0 = _mm512_extractf32x4_ps(a, 0);
        __m128 lane1 = _mm512_extractf32x4_ps(a, 1);
        __m128 lane2 = _mm512_extractf32x4_ps(a, 2);
        __m128 lane3 = _mm512_extractf32x4_ps(a, 3);
        __m128 res = pmul(pmul(lane0, lane1), pmul(lane2, lane3));
        res = pmul(res, _mm_permute_ps(res, _MM_SHUFFLE(0, 0, 3, 2)));
        return pfirst(pmul(res, _mm_permute_ps(res, _MM_SHUFFLE(0, 0, 0, 1))));
#endif
    }
    template <> EIGEN_STRONG_INLINE double predux_mul<Packet8d>(const Packet8d& a)
    {
        __m256d lane0 = _mm512_extractf64x4_pd(a, 0);
        __m256d lane1 = _mm512_extractf64x4_pd(a, 1);
        __m256d res = pmul(lane0, lane1);
        res = pmul(res, _mm256_permute2f128_pd(res, res, 1));
        return pfirst(pmul(res, _mm256_shuffle_pd(res, res, 1)));
    }

    template <> EIGEN_STRONG_INLINE float predux_min<Packet16f>(const Packet16f& a)
    {
        __m128 lane0 = _mm512_extractf32x4_ps(a, 0);
        __m128 lane1 = _mm512_extractf32x4_ps(a, 1);
        __m128 lane2 = _mm512_extractf32x4_ps(a, 2);
        __m128 lane3 = _mm512_extractf32x4_ps(a, 3);
        __m128 res = _mm_min_ps(_mm_min_ps(lane0, lane1), _mm_min_ps(lane2, lane3));
        res = _mm_min_ps(res, _mm_permute_ps(res, _MM_SHUFFLE(0, 0, 3, 2)));
        return pfirst(_mm_min_ps(res, _mm_permute_ps(res, _MM_SHUFFLE(0, 0, 0, 1))));
    }
    template <> EIGEN_STRONG_INLINE double predux_min<Packet8d>(const Packet8d& a)
    {
        __m256d lane0 = _mm512_extractf64x4_pd(a, 0);
        __m256d lane1 = _mm512_extractf64x4_pd(a, 1);
        __m256d res = _mm256_min_pd(lane0, lane1);
        res = _mm256_min_pd(res, _mm256_permute2f128_pd(res, res, 1));
        return pfirst(_mm256_min_pd(res, _mm256_shuffle_pd(res, res, 1)));
    }

    template <> EIGEN_STRONG_INLINE float predux_max<Packet16f>(const Packet16f& a)
    {
        __m128 lane0 = _mm512_extractf32x4_ps(a, 0);
        __m128 lane1 = _mm512_extractf32x4_ps(a, 1);
        __m128 lane2 = _mm512_extractf32x4_ps(a, 2);
        __m128 lane3 = _mm512_extractf32x4_ps(a, 3);
        __m128 res = _mm_max_ps(_mm_max_ps(lane0, lane1), _mm_max_ps(lane2, lane3));
        res = _mm_max_ps(res, _mm_permute_ps(res, _MM_SHUFFLE(0, 0, 3, 2)));
        return pfirst(_mm_max_ps(res, _mm_permute_ps(res, _MM_SHUFFLE(0, 0, 0, 1))));
    }

    template <> EIGEN_STRONG_INLINE double predux_max<Packet8d>(const Packet8d& a)
    {
        __m256d lane0 = _mm512_extractf64x4_pd(a, 0);
        __m256d lane1 = _mm512_extractf64x4_pd(a, 1);
        __m256d res = _mm256_max_pd(lane0, lane1);
        res = _mm256_max_pd(res, _mm256_permute2f128_pd(res, res, 1));
        return pfirst(_mm256_max_pd(res, _mm256_shuffle_pd(res, res, 1)));
    }

    template <> EIGEN_STRONG_INLINE bool predux_any(const Packet16f& x)
    {
        Packet16i xi = _mm512_castps_si512(x);
        __mmask16 tmp = _mm512_test_epi32_mask(xi, xi);
        return !_mm512_kortestz(tmp, tmp);
    }

#define PACK_OUTPUT(OUTPUT, INPUT, INDEX, STRIDE) EIGEN_INSERT_8f_INTO_16f(OUTPUT[INDEX], INPUT[INDEX], INPUT[INDEX + STRIDE]);

    EIGEN_DEVICE_FUNC inline void ptranspose(PacketBlock<Packet16f, 16>& kernel)
    {
        __m512 T0 = _mm512_unpacklo_ps(kernel.packet[0], kernel.packet[1]);
        __m512 T1 = _mm512_unpackhi_ps(kernel.packet[0], kernel.packet[1]);
        __m512 T2 = _mm512_unpacklo_ps(kernel.packet[2], kernel.packet[3]);
        __m512 T3 = _mm512_unpackhi_ps(kernel.packet[2], kernel.packet[3]);
        __m512 T4 = _mm512_unpacklo_ps(kernel.packet[4], kernel.packet[5]);
        __m512 T5 = _mm512_unpackhi_ps(kernel.packet[4], kernel.packet[5]);
        __m512 T6 = _mm512_unpacklo_ps(kernel.packet[6], kernel.packet[7]);
        __m512 T7 = _mm512_unpackhi_ps(kernel.packet[6], kernel.packet[7]);
        __m512 T8 = _mm512_unpacklo_ps(kernel.packet[8], kernel.packet[9]);
        __m512 T9 = _mm512_unpackhi_ps(kernel.packet[8], kernel.packet[9]);
        __m512 T10 = _mm512_unpacklo_ps(kernel.packet[10], kernel.packet[11]);
        __m512 T11 = _mm512_unpackhi_ps(kernel.packet[10], kernel.packet[11]);
        __m512 T12 = _mm512_unpacklo_ps(kernel.packet[12], kernel.packet[13]);
        __m512 T13 = _mm512_unpackhi_ps(kernel.packet[12], kernel.packet[13]);
        __m512 T14 = _mm512_unpacklo_ps(kernel.packet[14], kernel.packet[15]);
        __m512 T15 = _mm512_unpackhi_ps(kernel.packet[14], kernel.packet[15]);
        __m512 S0 = _mm512_shuffle_ps(T0, T2, _MM_SHUFFLE(1, 0, 1, 0));
        __m512 S1 = _mm512_shuffle_ps(T0, T2, _MM_SHUFFLE(3, 2, 3, 2));
        __m512 S2 = _mm512_shuffle_ps(T1, T3, _MM_SHUFFLE(1, 0, 1, 0));
        __m512 S3 = _mm512_shuffle_ps(T1, T3, _MM_SHUFFLE(3, 2, 3, 2));
        __m512 S4 = _mm512_shuffle_ps(T4, T6, _MM_SHUFFLE(1, 0, 1, 0));
        __m512 S5 = _mm512_shuffle_ps(T4, T6, _MM_SHUFFLE(3, 2, 3, 2));
        __m512 S6 = _mm512_shuffle_ps(T5, T7, _MM_SHUFFLE(1, 0, 1, 0));
        __m512 S7 = _mm512_shuffle_ps(T5, T7, _MM_SHUFFLE(3, 2, 3, 2));
        __m512 S8 = _mm512_shuffle_ps(T8, T10, _MM_SHUFFLE(1, 0, 1, 0));
        __m512 S9 = _mm512_shuffle_ps(T8, T10, _MM_SHUFFLE(3, 2, 3, 2));
        __m512 S10 = _mm512_shuffle_ps(T9, T11, _MM_SHUFFLE(1, 0, 1, 0));
        __m512 S11 = _mm512_shuffle_ps(T9, T11, _MM_SHUFFLE(3, 2, 3, 2));
        __m512 S12 = _mm512_shuffle_ps(T12, T14, _MM_SHUFFLE(1, 0, 1, 0));
        __m512 S13 = _mm512_shuffle_ps(T12, T14, _MM_SHUFFLE(3, 2, 3, 2));
        __m512 S14 = _mm512_shuffle_ps(T13, T15, _MM_SHUFFLE(1, 0, 1, 0));
        __m512 S15 = _mm512_shuffle_ps(T13, T15, _MM_SHUFFLE(3, 2, 3, 2));

        EIGEN_EXTRACT_8f_FROM_16f(S0, S0);
        EIGEN_EXTRACT_8f_FROM_16f(S1, S1);
        EIGEN_EXTRACT_8f_FROM_16f(S2, S2);
        EIGEN_EXTRACT_8f_FROM_16f(S3, S3);
        EIGEN_EXTRACT_8f_FROM_16f(S4, S4);
        EIGEN_EXTRACT_8f_FROM_16f(S5, S5);
        EIGEN_EXTRACT_8f_FROM_16f(S6, S6);
        EIGEN_EXTRACT_8f_FROM_16f(S7, S7);
        EIGEN_EXTRACT_8f_FROM_16f(S8, S8);
        EIGEN_EXTRACT_8f_FROM_16f(S9, S9);
        EIGEN_EXTRACT_8f_FROM_16f(S10, S10);
        EIGEN_EXTRACT_8f_FROM_16f(S11, S11);
        EIGEN_EXTRACT_8f_FROM_16f(S12, S12);
        EIGEN_EXTRACT_8f_FROM_16f(S13, S13);
        EIGEN_EXTRACT_8f_FROM_16f(S14, S14);
        EIGEN_EXTRACT_8f_FROM_16f(S15, S15);

        PacketBlock<Packet8f, 32> tmp;

        tmp.packet[0] = _mm256_permute2f128_ps(S0_0, S4_0, 0x20);
        tmp.packet[1] = _mm256_permute2f128_ps(S1_0, S5_0, 0x20);
        tmp.packet[2] = _mm256_permute2f128_ps(S2_0, S6_0, 0x20);
        tmp.packet[3] = _mm256_permute2f128_ps(S3_0, S7_0, 0x20);
        tmp.packet[4] = _mm256_permute2f128_ps(S0_0, S4_0, 0x31);
        tmp.packet[5] = _mm256_permute2f128_ps(S1_0, S5_0, 0x31);
        tmp.packet[6] = _mm256_permute2f128_ps(S2_0, S6_0, 0x31);
        tmp.packet[7] = _mm256_permute2f128_ps(S3_0, S7_0, 0x31);

        tmp.packet[8] = _mm256_permute2f128_ps(S0_1, S4_1, 0x20);
        tmp.packet[9] = _mm256_permute2f128_ps(S1_1, S5_1, 0x20);
        tmp.packet[10] = _mm256_permute2f128_ps(S2_1, S6_1, 0x20);
        tmp.packet[11] = _mm256_permute2f128_ps(S3_1, S7_1, 0x20);
        tmp.packet[12] = _mm256_permute2f128_ps(S0_1, S4_1, 0x31);
        tmp.packet[13] = _mm256_permute2f128_ps(S1_1, S5_1, 0x31);
        tmp.packet[14] = _mm256_permute2f128_ps(S2_1, S6_1, 0x31);
        tmp.packet[15] = _mm256_permute2f128_ps(S3_1, S7_1, 0x31);

        // Second set of _m256 outputs
        tmp.packet[16] = _mm256_permute2f128_ps(S8_0, S12_0, 0x20);
        tmp.packet[17] = _mm256_permute2f128_ps(S9_0, S13_0, 0x20);
        tmp.packet[18] = _mm256_permute2f128_ps(S10_0, S14_0, 0x20);
        tmp.packet[19] = _mm256_permute2f128_ps(S11_0, S15_0, 0x20);
        tmp.packet[20] = _mm256_permute2f128_ps(S8_0, S12_0, 0x31);
        tmp.packet[21] = _mm256_permute2f128_ps(S9_0, S13_0, 0x31);
        tmp.packet[22] = _mm256_permute2f128_ps(S10_0, S14_0, 0x31);
        tmp.packet[23] = _mm256_permute2f128_ps(S11_0, S15_0, 0x31);

        tmp.packet[24] = _mm256_permute2f128_ps(S8_1, S12_1, 0x20);
        tmp.packet[25] = _mm256_permute2f128_ps(S9_1, S13_1, 0x20);
        tmp.packet[26] = _mm256_permute2f128_ps(S10_1, S14_1, 0x20);
        tmp.packet[27] = _mm256_permute2f128_ps(S11_1, S15_1, 0x20);
        tmp.packet[28] = _mm256_permute2f128_ps(S8_1, S12_1, 0x31);
        tmp.packet[29] = _mm256_permute2f128_ps(S9_1, S13_1, 0x31);
        tmp.packet[30] = _mm256_permute2f128_ps(S10_1, S14_1, 0x31);
        tmp.packet[31] = _mm256_permute2f128_ps(S11_1, S15_1, 0x31);

        // Pack them into the output
        PACK_OUTPUT(kernel.packet, tmp.packet, 0, 16);
        PACK_OUTPUT(kernel.packet, tmp.packet, 1, 16);
        PACK_OUTPUT(kernel.packet, tmp.packet, 2, 16);
        PACK_OUTPUT(kernel.packet, tmp.packet, 3, 16);

        PACK_OUTPUT(kernel.packet, tmp.packet, 4, 16);
        PACK_OUTPUT(kernel.packet, tmp.packet, 5, 16);
        PACK_OUTPUT(kernel.packet, tmp.packet, 6, 16);
        PACK_OUTPUT(kernel.packet, tmp.packet, 7, 16);

        PACK_OUTPUT(kernel.packet, tmp.packet, 8, 16);
        PACK_OUTPUT(kernel.packet, tmp.packet, 9, 16);
        PACK_OUTPUT(kernel.packet, tmp.packet, 10, 16);
        PACK_OUTPUT(kernel.packet, tmp.packet, 11, 16);

        PACK_OUTPUT(kernel.packet, tmp.packet, 12, 16);
        PACK_OUTPUT(kernel.packet, tmp.packet, 13, 16);
        PACK_OUTPUT(kernel.packet, tmp.packet, 14, 16);
        PACK_OUTPUT(kernel.packet, tmp.packet, 15, 16);
    }
#define PACK_OUTPUT_2(OUTPUT, INPUT, INDEX, STRIDE) EIGEN_INSERT_8f_INTO_16f(OUTPUT[INDEX], INPUT[2 * INDEX], INPUT[2 * INDEX + STRIDE]);

    EIGEN_DEVICE_FUNC inline void ptranspose(PacketBlock<Packet16f, 4>& kernel)
    {
        __m512 T0 = _mm512_unpacklo_ps(kernel.packet[0], kernel.packet[1]);
        __m512 T1 = _mm512_unpackhi_ps(kernel.packet[0], kernel.packet[1]);
        __m512 T2 = _mm512_unpacklo_ps(kernel.packet[2], kernel.packet[3]);
        __m512 T3 = _mm512_unpackhi_ps(kernel.packet[2], kernel.packet[3]);

        __m512 S0 = _mm512_shuffle_ps(T0, T2, _MM_SHUFFLE(1, 0, 1, 0));
        __m512 S1 = _mm512_shuffle_ps(T0, T2, _MM_SHUFFLE(3, 2, 3, 2));
        __m512 S2 = _mm512_shuffle_ps(T1, T3, _MM_SHUFFLE(1, 0, 1, 0));
        __m512 S3 = _mm512_shuffle_ps(T1, T3, _MM_SHUFFLE(3, 2, 3, 2));

        EIGEN_EXTRACT_8f_FROM_16f(S0, S0);
        EIGEN_EXTRACT_8f_FROM_16f(S1, S1);
        EIGEN_EXTRACT_8f_FROM_16f(S2, S2);
        EIGEN_EXTRACT_8f_FROM_16f(S3, S3);

        PacketBlock<Packet8f, 8> tmp;

        tmp.packet[0] = _mm256_permute2f128_ps(S0_0, S1_0, 0x20);
        tmp.packet[1] = _mm256_permute2f128_ps(S2_0, S3_0, 0x20);
        tmp.packet[2] = _mm256_permute2f128_ps(S0_0, S1_0, 0x31);
        tmp.packet[3] = _mm256_permute2f128_ps(S2_0, S3_0, 0x31);

        tmp.packet[4] = _mm256_permute2f128_ps(S0_1, S1_1, 0x20);
        tmp.packet[5] = _mm256_permute2f128_ps(S2_1, S3_1, 0x20);
        tmp.packet[6] = _mm256_permute2f128_ps(S0_1, S1_1, 0x31);
        tmp.packet[7] = _mm256_permute2f128_ps(S2_1, S3_1, 0x31);

        PACK_OUTPUT_2(kernel.packet, tmp.packet, 0, 1);
        PACK_OUTPUT_2(kernel.packet, tmp.packet, 1, 1);
        PACK_OUTPUT_2(kernel.packet, tmp.packet, 2, 1);
        PACK_OUTPUT_2(kernel.packet, tmp.packet, 3, 1);
    }

#define PACK_OUTPUT_SQ_D(OUTPUT, INPUT, INDEX, STRIDE)                  \
    OUTPUT[INDEX] = _mm512_insertf64x4(OUTPUT[INDEX], INPUT[INDEX], 0); \
    OUTPUT[INDEX] = _mm512_insertf64x4(OUTPUT[INDEX], INPUT[INDEX + STRIDE], 1);

#define PACK_OUTPUT_D(OUTPUT, INPUT, INDEX, STRIDE)                           \
    OUTPUT[INDEX] = _mm512_insertf64x4(OUTPUT[INDEX], INPUT[(2 * INDEX)], 0); \
    OUTPUT[INDEX] = _mm512_insertf64x4(OUTPUT[INDEX], INPUT[(2 * INDEX) + STRIDE], 1);

    EIGEN_DEVICE_FUNC inline void ptranspose(PacketBlock<Packet8d, 4>& kernel)
    {
        __m512d T0 = _mm512_shuffle_pd(kernel.packet[0], kernel.packet[1], 0);
        __m512d T1 = _mm512_shuffle_pd(kernel.packet[0], kernel.packet[1], 0xff);
        __m512d T2 = _mm512_shuffle_pd(kernel.packet[2], kernel.packet[3], 0);
        __m512d T3 = _mm512_shuffle_pd(kernel.packet[2], kernel.packet[3], 0xff);

        PacketBlock<Packet4d, 8> tmp;

        tmp.packet[0] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T0, 0), _mm512_extractf64x4_pd(T2, 0), 0x20);
        tmp.packet[1] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T1, 0), _mm512_extractf64x4_pd(T3, 0), 0x20);
        tmp.packet[2] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T0, 0), _mm512_extractf64x4_pd(T2, 0), 0x31);
        tmp.packet[3] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T1, 0), _mm512_extractf64x4_pd(T3, 0), 0x31);

        tmp.packet[4] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T0, 1), _mm512_extractf64x4_pd(T2, 1), 0x20);
        tmp.packet[5] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T1, 1), _mm512_extractf64x4_pd(T3, 1), 0x20);
        tmp.packet[6] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T0, 1), _mm512_extractf64x4_pd(T2, 1), 0x31);
        tmp.packet[7] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T1, 1), _mm512_extractf64x4_pd(T3, 1), 0x31);

        PACK_OUTPUT_D(kernel.packet, tmp.packet, 0, 1);
        PACK_OUTPUT_D(kernel.packet, tmp.packet, 1, 1);
        PACK_OUTPUT_D(kernel.packet, tmp.packet, 2, 1);
        PACK_OUTPUT_D(kernel.packet, tmp.packet, 3, 1);
    }

    EIGEN_DEVICE_FUNC inline void ptranspose(PacketBlock<Packet8d, 8>& kernel)
    {
        __m512d T0 = _mm512_unpacklo_pd(kernel.packet[0], kernel.packet[1]);
        __m512d T1 = _mm512_unpackhi_pd(kernel.packet[0], kernel.packet[1]);
        __m512d T2 = _mm512_unpacklo_pd(kernel.packet[2], kernel.packet[3]);
        __m512d T3 = _mm512_unpackhi_pd(kernel.packet[2], kernel.packet[3]);
        __m512d T4 = _mm512_unpacklo_pd(kernel.packet[4], kernel.packet[5]);
        __m512d T5 = _mm512_unpackhi_pd(kernel.packet[4], kernel.packet[5]);
        __m512d T6 = _mm512_unpacklo_pd(kernel.packet[6], kernel.packet[7]);
        __m512d T7 = _mm512_unpackhi_pd(kernel.packet[6], kernel.packet[7]);

        PacketBlock<Packet4d, 16> tmp;

        tmp.packet[0] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T0, 0), _mm512_extractf64x4_pd(T2, 0), 0x20);
        tmp.packet[1] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T1, 0), _mm512_extractf64x4_pd(T3, 0), 0x20);
        tmp.packet[2] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T0, 0), _mm512_extractf64x4_pd(T2, 0), 0x31);
        tmp.packet[3] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T1, 0), _mm512_extractf64x4_pd(T3, 0), 0x31);

        tmp.packet[4] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T0, 1), _mm512_extractf64x4_pd(T2, 1), 0x20);
        tmp.packet[5] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T1, 1), _mm512_extractf64x4_pd(T3, 1), 0x20);
        tmp.packet[6] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T0, 1), _mm512_extractf64x4_pd(T2, 1), 0x31);
        tmp.packet[7] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T1, 1), _mm512_extractf64x4_pd(T3, 1), 0x31);

        tmp.packet[8] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T4, 0), _mm512_extractf64x4_pd(T6, 0), 0x20);
        tmp.packet[9] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T5, 0), _mm512_extractf64x4_pd(T7, 0), 0x20);
        tmp.packet[10] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T4, 0), _mm512_extractf64x4_pd(T6, 0), 0x31);
        tmp.packet[11] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T5, 0), _mm512_extractf64x4_pd(T7, 0), 0x31);

        tmp.packet[12] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T4, 1), _mm512_extractf64x4_pd(T6, 1), 0x20);
        tmp.packet[13] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T5, 1), _mm512_extractf64x4_pd(T7, 1), 0x20);
        tmp.packet[14] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T4, 1), _mm512_extractf64x4_pd(T6, 1), 0x31);
        tmp.packet[15] = _mm256_permute2f128_pd(_mm512_extractf64x4_pd(T5, 1), _mm512_extractf64x4_pd(T7, 1), 0x31);

        PACK_OUTPUT_SQ_D(kernel.packet, tmp.packet, 0, 8);
        PACK_OUTPUT_SQ_D(kernel.packet, tmp.packet, 1, 8);
        PACK_OUTPUT_SQ_D(kernel.packet, tmp.packet, 2, 8);
        PACK_OUTPUT_SQ_D(kernel.packet, tmp.packet, 3, 8);

        PACK_OUTPUT_SQ_D(kernel.packet, tmp.packet, 4, 8);
        PACK_OUTPUT_SQ_D(kernel.packet, tmp.packet, 5, 8);
        PACK_OUTPUT_SQ_D(kernel.packet, tmp.packet, 6, 8);
        PACK_OUTPUT_SQ_D(kernel.packet, tmp.packet, 7, 8);
    }
    template <> EIGEN_STRONG_INLINE Packet16f pblend(const Selector<16>& /*ifPacket*/, const Packet16f& /*thenPacket*/, const Packet16f& /*elsePacket*/)
    {
        assert(false && "To be implemented");
        return Packet16f();
    }
    template <> EIGEN_STRONG_INLINE Packet8d pblend(const Selector<8>& ifPacket, const Packet8d& thenPacket, const Packet8d& elsePacket)
    {
        __mmask8 m = (ifPacket.select[0]) | (ifPacket.select[1] << 1) | (ifPacket.select[2] << 2) | (ifPacket.select[3] << 3) | (ifPacket.select[4] << 4) |
                     (ifPacket.select[5] << 5) | (ifPacket.select[6] << 6) | (ifPacket.select[7] << 7);
        return _mm512_mask_blend_pd(m, elsePacket, thenPacket);
    }

    // Packet math for Eigen::half
    template <> EIGEN_STRONG_INLINE Packet16h pset1<Packet16h>(const Eigen::half& from) { return _mm256_set1_epi16(from.x); }

    template <> EIGEN_STRONG_INLINE Eigen::half pfirst<Packet16h>(const Packet16h& from)
    {
        return half_impl::raw_uint16_to_half(static_cast<unsigned short>(_mm256_extract_epi16(from, 0)));
    }

    template <> EIGEN_STRONG_INLINE Packet16h pload<Packet16h>(const Eigen::half* from) { return _mm256_load_si256(reinterpret_cast<const __m256i*>(from)); }

    template <> EIGEN_STRONG_INLINE Packet16h ploadu<Packet16h>(const Eigen::half* from) { return _mm256_loadu_si256(reinterpret_cast<const __m256i*>(from)); }

    template <> EIGEN_STRONG_INLINE void pstore<half>(Eigen::half* to, const Packet16h& from)
    {
        // (void*) -> workaround clang warning:
        // cast from 'Eigen::half *' to '__m256i *' increases required alignment from 2 to 32
        _mm256_store_si256((__m256i*)(void*)to, from);
    }

    template <> EIGEN_STRONG_INLINE void pstoreu<half>(Eigen::half* to, const Packet16h& from)
    {
        // (void*) -> workaround clang warning:
        // cast from 'Eigen::half *' to '__m256i *' increases required alignment from 2 to 32
        _mm256_storeu_si256((__m256i*)(void*)to, from);
    }

    template <> EIGEN_STRONG_INLINE Packet16h ploaddup<Packet16h>(const Eigen::half* from)
    {
        unsigned short a = from[0].x;
        unsigned short b = from[1].x;
        unsigned short c = from[2].x;
        unsigned short d = from[3].x;
        unsigned short e = from[4].x;
        unsigned short f = from[5].x;
        unsigned short g = from[6].x;
        unsigned short h = from[7].x;
        return _mm256_set_epi16(h, h, g, g, f, f, e, e, d, d, c, c, b, b, a, a);
    }

    template <> EIGEN_STRONG_INLINE Packet16h ploadquad(const Eigen::half* from)
    {
        unsigned short a = from[0].x;
        unsigned short b = from[1].x;
        unsigned short c = from[2].x;
        unsigned short d = from[3].x;
        return _mm256_set_epi16(d, d, d, d, c, c, c, c, b, b, b, b, a, a, a, a);
    }

    EIGEN_STRONG_INLINE Packet16f half2float(const Packet16h& a)
    {
#ifdef EIGEN_HAS_FP16_C
        return _mm512_cvtph_ps(a);
#else
        EIGEN_ALIGN64 half aux[16];
        pstore(aux, a);
        float f0(aux[0]);
        float f1(aux[1]);
        float f2(aux[2]);
        float f3(aux[3]);
        float f4(aux[4]);
        float f5(aux[5]);
        float f6(aux[6]);
        float f7(aux[7]);
        float f8(aux[8]);
        float f9(aux[9]);
        float fa(aux[10]);
        float fb(aux[11]);
        float fc(aux[12]);
        float fd(aux[13]);
        float fe(aux[14]);
        float ff(aux[15]);

        return _mm512_set_ps(ff, fe, fd, fc, fb, fa, f9, f8, f7, f6, f5, f4, f3, f2, f1, f0);
#endif
    }

    EIGEN_STRONG_INLINE Packet16h float2half(const Packet16f& a)
    {
#ifdef EIGEN_HAS_FP16_C
        return _mm512_cvtps_ph(a, _MM_FROUND_TO_NEAREST_INT | _MM_FROUND_NO_EXC);
#else
        EIGEN_ALIGN64 float aux[16];
        pstore(aux, a);
        half h0(aux[0]);
        half h1(aux[1]);
        half h2(aux[2]);
        half h3(aux[3]);
        half h4(aux[4]);
        half h5(aux[5]);
        half h6(aux[6]);
        half h7(aux[7]);
        half h8(aux[8]);
        half h9(aux[9]);
        half ha(aux[10]);
        half hb(aux[11]);
        half hc(aux[12]);
        half hd(aux[13]);
        half he(aux[14]);
        half hf(aux[15]);

        return _mm256_set_epi16(hf.x, he.x, hd.x, hc.x, hb.x, ha.x, h9.x, h8.x, h7.x, h6.x, h5.x, h4.x, h3.x, h2.x, h1.x, h0.x);
#endif
    }

    template <> EIGEN_STRONG_INLINE Packet16h ptrue(const Packet16h& a) { return ptrue(Packet8i(a)); }

    template <> EIGEN_STRONG_INLINE Packet16h pabs(const Packet16h& a)
    {
        const __m256i sign_mask = _mm256_set1_epi16(static_cast<numext::uint16_t>(0x8000));
        return _mm256_andnot_si256(sign_mask, a);
    }

    template <> EIGEN_STRONG_INLINE Packet16h pmin<Packet16h>(const Packet16h& a, const Packet16h& b)
    {
        return float2half(pmin<Packet16f>(half2float(a), half2float(b)));
    }

    template <> EIGEN_STRONG_INLINE Packet16h pmax<Packet16h>(const Packet16h& a, const Packet16h& b)
    {
        return float2half(pmax<Packet16f>(half2float(a), half2float(b)));
    }

    template <> EIGEN_STRONG_INLINE Packet16h plset<Packet16h>(const half& a) { return float2half(plset<Packet16f>(static_cast<float>(a))); }

    template <> EIGEN_STRONG_INLINE Packet16h por(const Packet16h& a, const Packet16h& b)
    {
        // in some cases Packet8i is a wrapper around __m256i, so we need to
        // cast to Packet8i to call the correct overload.
        return por(Packet8i(a), Packet8i(b));
    }
    template <> EIGEN_STRONG_INLINE Packet16h pxor(const Packet16h& a, const Packet16h& b) { return pxor(Packet8i(a), Packet8i(b)); }
    template <> EIGEN_STRONG_INLINE Packet16h pand(const Packet16h& a, const Packet16h& b) { return pand(Packet8i(a), Packet8i(b)); }
    template <> EIGEN_STRONG_INLINE Packet16h pandnot(const Packet16h& a, const Packet16h& b) { return pandnot(Packet8i(a), Packet8i(b)); }

    template <> EIGEN_STRONG_INLINE Packet16h pselect(const Packet16h& mask, const Packet16h& a, const Packet16h& b) { return _mm256_blendv_epi8(b, a, mask); }

    template <> EIGEN_STRONG_INLINE Packet16h pround<Packet16h>(const Packet16h& a) { return float2half(pround<Packet16f>(half2float(a))); }

    template <> EIGEN_STRONG_INLINE Packet16h print<Packet16h>(const Packet16h& a) { return float2half(print<Packet16f>(half2float(a))); }

    template <> EIGEN_STRONG_INLINE Packet16h pceil<Packet16h>(const Packet16h& a) { return float2half(pceil<Packet16f>(half2float(a))); }

    template <> EIGEN_STRONG_INLINE Packet16h pfloor<Packet16h>(const Packet16h& a) { return float2half(pfloor<Packet16f>(half2float(a))); }

    template <> EIGEN_STRONG_INLINE Packet16h pcmp_eq(const Packet16h& a, const Packet16h& b)
    {
        Packet16f af = half2float(a);
        Packet16f bf = half2float(b);
        return Pack32To16(pcmp_eq(af, bf));
    }

    template <> EIGEN_STRONG_INLINE Packet16h pcmp_le(const Packet16h& a, const Packet16h& b) { return Pack32To16(pcmp_le(half2float(a), half2float(b))); }

    template <> EIGEN_STRONG_INLINE Packet16h pcmp_lt(const Packet16h& a, const Packet16h& b) { return Pack32To16(pcmp_lt(half2float(a), half2float(b))); }

    template <> EIGEN_STRONG_INLINE Packet16h pcmp_lt_or_nan(const Packet16h& a, const Packet16h& b)
    {
        return Pack32To16(pcmp_lt_or_nan(half2float(a), half2float(b)));
    }

    template <> EIGEN_STRONG_INLINE Packet16h pconj(const Packet16h& a) { return a; }

    template <> EIGEN_STRONG_INLINE Packet16h pnegate(const Packet16h& a)
    {
        Packet16h sign_mask = _mm256_set1_epi16(static_cast<unsigned short>(0x8000));
        return _mm256_xor_si256(a, sign_mask);
    }

    template <> EIGEN_STRONG_INLINE Packet16h padd<Packet16h>(const Packet16h& a, const Packet16h& b)
    {
        Packet16f af = half2float(a);
        Packet16f bf = half2float(b);
        Packet16f rf = padd(af, bf);
        return float2half(rf);
    }

    template <> EIGEN_STRONG_INLINE Packet16h psub<Packet16h>(const Packet16h& a, const Packet16h& b)
    {
        Packet16f af = half2float(a);
        Packet16f bf = half2float(b);
        Packet16f rf = psub(af, bf);
        return float2half(rf);
    }

    template <> EIGEN_STRONG_INLINE Packet16h pmul<Packet16h>(const Packet16h& a, const Packet16h& b)
    {
        Packet16f af = half2float(a);
        Packet16f bf = half2float(b);
        Packet16f rf = pmul(af, bf);
        return float2half(rf);
    }

    template <> EIGEN_STRONG_INLINE Packet16h pdiv<Packet16h>(const Packet16h& a, const Packet16h& b)
    {
        Packet16f af = half2float(a);
        Packet16f bf = half2float(b);
        Packet16f rf = pdiv(af, bf);
        return float2half(rf);
    }

    template <> EIGEN_STRONG_INLINE half predux<Packet16h>(const Packet16h& from)
    {
        Packet16f from_float = half2float(from);
        return half(predux(from_float));
    }

    template <> EIGEN_STRONG_INLINE Packet8h predux_half_dowto4<Packet16h>(const Packet16h& a)
    {
        Packet8h lane0 = _mm256_extractf128_si256(a, 0);
        Packet8h lane1 = _mm256_extractf128_si256(a, 1);
        return padd<Packet8h>(lane0, lane1);
    }

    template <> EIGEN_STRONG_INLINE Eigen::half predux_max<Packet16h>(const Packet16h& a)
    {
        Packet16f af = half2float(a);
        float reduced = predux_max<Packet16f>(af);
        return Eigen::half(reduced);
    }

    template <> EIGEN_STRONG_INLINE Eigen::half predux_min<Packet16h>(const Packet16h& a)
    {
        Packet16f af = half2float(a);
        float reduced = predux_min<Packet16f>(af);
        return Eigen::half(reduced);
    }

    template <> EIGEN_STRONG_INLINE half predux_mul<Packet16h>(const Packet16h& from)
    {
        Packet16f from_float = half2float(from);
        return half(predux_mul(from_float));
    }

    template <> EIGEN_STRONG_INLINE Packet16h preverse(const Packet16h& a)
    {
        __m128i m = _mm_setr_epi8(14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1);
        return _mm256_insertf128_si256(
            _mm256_castsi128_si256(_mm_shuffle_epi8(_mm256_extractf128_si256(a, 1), m)), _mm_shuffle_epi8(_mm256_extractf128_si256(a, 0), m), 1);
    }

    template <> EIGEN_STRONG_INLINE Packet16h pgather<Eigen::half, Packet16h>(const Eigen::half* from, Index stride)
    {
        return _mm256_set_epi16(from[15 * stride].x,
                                from[14 * stride].x,
                                from[13 * stride].x,
                                from[12 * stride].x,
                                from[11 * stride].x,
                                from[10 * stride].x,
                                from[9 * stride].x,
                                from[8 * stride].x,
                                from[7 * stride].x,
                                from[6 * stride].x,
                                from[5 * stride].x,
                                from[4 * stride].x,
                                from[3 * stride].x,
                                from[2 * stride].x,
                                from[1 * stride].x,
                                from[0 * stride].x);
    }

    template <> EIGEN_STRONG_INLINE void pscatter<half, Packet16h>(half* to, const Packet16h& from, Index stride)
    {
        EIGEN_ALIGN64 half aux[16];
        pstore(aux, from);
        to[stride * 0] = aux[0];
        to[stride * 1] = aux[1];
        to[stride * 2] = aux[2];
        to[stride * 3] = aux[3];
        to[stride * 4] = aux[4];
        to[stride * 5] = aux[5];
        to[stride * 6] = aux[6];
        to[stride * 7] = aux[7];
        to[stride * 8] = aux[8];
        to[stride * 9] = aux[9];
        to[stride * 10] = aux[10];
        to[stride * 11] = aux[11];
        to[stride * 12] = aux[12];
        to[stride * 13] = aux[13];
        to[stride * 14] = aux[14];
        to[stride * 15] = aux[15];
    }

    EIGEN_STRONG_INLINE void ptranspose(PacketBlock<Packet16h, 16>& kernel)
    {
        __m256i a = kernel.packet[0];
        __m256i b = kernel.packet[1];
        __m256i c = kernel.packet[2];
        __m256i d = kernel.packet[3];
        __m256i e = kernel.packet[4];
        __m256i f = kernel.packet[5];
        __m256i g = kernel.packet[6];
        __m256i h = kernel.packet[7];
        __m256i i = kernel.packet[8];
        __m256i j = kernel.packet[9];
        __m256i k = kernel.packet[10];
        __m256i l = kernel.packet[11];
        __m256i m = kernel.packet[12];
        __m256i n = kernel.packet[13];
        __m256i o = kernel.packet[14];
        __m256i p = kernel.packet[15];

        __m256i ab_07 = _mm256_unpacklo_epi16(a, b);
        __m256i cd_07 = _mm256_unpacklo_epi16(c, d);
        __m256i ef_07 = _mm256_unpacklo_epi16(e, f);
        __m256i gh_07 = _mm256_unpacklo_epi16(g, h);
        __m256i ij_07 = _mm256_unpacklo_epi16(i, j);
        __m256i kl_07 = _mm256_unpacklo_epi16(k, l);
        __m256i mn_07 = _mm256_unpacklo_epi16(m, n);
        __m256i op_07 = _mm256_unpacklo_epi16(o, p);

        __m256i ab_8f = _mm256_unpackhi_epi16(a, b);
        __m256i cd_8f = _mm256_unpackhi_epi16(c, d);
        __m256i ef_8f = _mm256_unpackhi_epi16(e, f);
        __m256i gh_8f = _mm256_unpackhi_epi16(g, h);
        __m256i ij_8f = _mm256_unpackhi_epi16(i, j);
        __m256i kl_8f = _mm256_unpackhi_epi16(k, l);
        __m256i mn_8f = _mm256_unpackhi_epi16(m, n);
        __m256i op_8f = _mm256_unpackhi_epi16(o, p);

        __m256i abcd_03 = _mm256_unpacklo_epi32(ab_07, cd_07);
        __m256i abcd_47 = _mm256_unpackhi_epi32(ab_07, cd_07);
        __m256i efgh_03 = _mm256_unpacklo_epi32(ef_07, gh_07);
        __m256i efgh_47 = _mm256_unpackhi_epi32(ef_07, gh_07);
        __m256i ijkl_03 = _mm256_unpacklo_epi32(ij_07, kl_07);
        __m256i ijkl_47 = _mm256_unpackhi_epi32(ij_07, kl_07);
        __m256i mnop_03 = _mm256_unpacklo_epi32(mn_07, op_07);
        __m256i mnop_47 = _mm256_unpackhi_epi32(mn_07, op_07);

        __m256i abcd_8b = _mm256_unpacklo_epi32(ab_8f, cd_8f);
        __m256i abcd_cf = _mm256_unpackhi_epi32(ab_8f, cd_8f);
        __m256i efgh_8b = _mm256_unpacklo_epi32(ef_8f, gh_8f);
        __m256i efgh_cf = _mm256_unpackhi_epi32(ef_8f, gh_8f);
        __m256i ijkl_8b = _mm256_unpacklo_epi32(ij_8f, kl_8f);
        __m256i ijkl_cf = _mm256_unpackhi_epi32(ij_8f, kl_8f);
        __m256i mnop_8b = _mm256_unpacklo_epi32(mn_8f, op_8f);
        __m256i mnop_cf = _mm256_unpackhi_epi32(mn_8f, op_8f);

        __m256i abcdefgh_01 = _mm256_unpacklo_epi64(abcd_03, efgh_03);
        __m256i abcdefgh_23 = _mm256_unpackhi_epi64(abcd_03, efgh_03);
        __m256i ijklmnop_01 = _mm256_unpacklo_epi64(ijkl_03, mnop_03);
        __m256i ijklmnop_23 = _mm256_unpackhi_epi64(ijkl_03, mnop_03);
        __m256i abcdefgh_45 = _mm256_unpacklo_epi64(abcd_47, efgh_47);
        __m256i abcdefgh_67 = _mm256_unpackhi_epi64(abcd_47, efgh_47);
        __m256i ijklmnop_45 = _mm256_unpacklo_epi64(ijkl_47, mnop_47);
        __m256i ijklmnop_67 = _mm256_unpackhi_epi64(ijkl_47, mnop_47);
        __m256i abcdefgh_89 = _mm256_unpacklo_epi64(abcd_8b, efgh_8b);
        __m256i abcdefgh_ab = _mm256_unpackhi_epi64(abcd_8b, efgh_8b);
        __m256i ijklmnop_89 = _mm256_unpacklo_epi64(ijkl_8b, mnop_8b);
        __m256i ijklmnop_ab = _mm256_unpackhi_epi64(ijkl_8b, mnop_8b);
        __m256i abcdefgh_cd = _mm256_unpacklo_epi64(abcd_cf, efgh_cf);
        __m256i abcdefgh_ef = _mm256_unpackhi_epi64(abcd_cf, efgh_cf);
        __m256i ijklmnop_cd = _mm256_unpacklo_epi64(ijkl_cf, mnop_cf);
        __m256i ijklmnop_ef = _mm256_unpackhi_epi64(ijkl_cf, mnop_cf);

        // NOTE: no unpacklo/hi instr in this case, so using permute instr.
        __m256i a_p_0 = _mm256_permute2x128_si256(abcdefgh_01, ijklmnop_01, 0x20);
        __m256i a_p_1 = _mm256_permute2x128_si256(abcdefgh_23, ijklmnop_23, 0x20);
        __m256i a_p_2 = _mm256_permute2x128_si256(abcdefgh_45, ijklmnop_45, 0x20);
        __m256i a_p_3 = _mm256_permute2x128_si256(abcdefgh_67, ijklmnop_67, 0x20);
        __m256i a_p_4 = _mm256_permute2x128_si256(abcdefgh_89, ijklmnop_89, 0x20);
        __m256i a_p_5 = _mm256_permute2x128_si256(abcdefgh_ab, ijklmnop_ab, 0x20);
        __m256i a_p_6 = _mm256_permute2x128_si256(abcdefgh_cd, ijklmnop_cd, 0x20);
        __m256i a_p_7 = _mm256_permute2x128_si256(abcdefgh_ef, ijklmnop_ef, 0x20);
        __m256i a_p_8 = _mm256_permute2x128_si256(abcdefgh_01, ijklmnop_01, 0x31);
        __m256i a_p_9 = _mm256_permute2x128_si256(abcdefgh_23, ijklmnop_23, 0x31);
        __m256i a_p_a = _mm256_permute2x128_si256(abcdefgh_45, ijklmnop_45, 0x31);
        __m256i a_p_b = _mm256_permute2x128_si256(abcdefgh_67, ijklmnop_67, 0x31);
        __m256i a_p_c = _mm256_permute2x128_si256(abcdefgh_89, ijklmnop_89, 0x31);
        __m256i a_p_d = _mm256_permute2x128_si256(abcdefgh_ab, ijklmnop_ab, 0x31);
        __m256i a_p_e = _mm256_permute2x128_si256(abcdefgh_cd, ijklmnop_cd, 0x31);
        __m256i a_p_f = _mm256_permute2x128_si256(abcdefgh_ef, ijklmnop_ef, 0x31);

        kernel.packet[0] = a_p_0;
        kernel.packet[1] = a_p_1;
        kernel.packet[2] = a_p_2;
        kernel.packet[3] = a_p_3;
        kernel.packet[4] = a_p_4;
        kernel.packet[5] = a_p_5;
        kernel.packet[6] = a_p_6;
        kernel.packet[7] = a_p_7;
        kernel.packet[8] = a_p_8;
        kernel.packet[9] = a_p_9;
        kernel.packet[10] = a_p_a;
        kernel.packet[11] = a_p_b;
        kernel.packet[12] = a_p_c;
        kernel.packet[13] = a_p_d;
        kernel.packet[14] = a_p_e;
        kernel.packet[15] = a_p_f;
    }

    EIGEN_STRONG_INLINE void ptranspose(PacketBlock<Packet16h, 8>& kernel)
    {
        EIGEN_ALIGN64 half in[8][16];
        pstore<half>(in[0], kernel.packet[0]);
        pstore<half>(in[1], kernel.packet[1]);
        pstore<half>(in[2], kernel.packet[2]);
        pstore<half>(in[3], kernel.packet[3]);
        pstore<half>(in[4], kernel.packet[4]);
        pstore<half>(in[5], kernel.packet[5]);
        pstore<half>(in[6], kernel.packet[6]);
        pstore<half>(in[7], kernel.packet[7]);

        EIGEN_ALIGN64 half out[8][16];

        for (int i = 0; i < 8; ++i)
        {
            for (int j = 0; j < 8; ++j) { out[i][j] = in[j][2 * i]; }
            for (int j = 0; j < 8; ++j) { out[i][j + 8] = in[j][2 * i + 1]; }
        }

        kernel.packet[0] = pload<Packet16h>(out[0]);
        kernel.packet[1] = pload<Packet16h>(out[1]);
        kernel.packet[2] = pload<Packet16h>(out[2]);
        kernel.packet[3] = pload<Packet16h>(out[3]);
        kernel.packet[4] = pload<Packet16h>(out[4]);
        kernel.packet[5] = pload<Packet16h>(out[5]);
        kernel.packet[6] = pload<Packet16h>(out[6]);
        kernel.packet[7] = pload<Packet16h>(out[7]);
    }

    EIGEN_STRONG_INLINE void ptranspose(PacketBlock<Packet16h, 4>& kernel)
    {
        EIGEN_ALIGN64 half in[4][16];
        pstore<half>(in[0], kernel.packet[0]);
        pstore<half>(in[1], kernel.packet[1]);
        pstore<half>(in[2], kernel.packet[2]);
        pstore<half>(in[3], kernel.packet[3]);

        EIGEN_ALIGN64 half out[4][16];

        for (int i = 0; i < 4; ++i)
        {
            for (int j = 0; j < 4; ++j) { out[i][j] = in[j][4 * i]; }
            for (int j = 0; j < 4; ++j) { out[i][j + 4] = in[j][4 * i + 1]; }
            for (int j = 0; j < 4; ++j) { out[i][j + 8] = in[j][4 * i + 2]; }
            for (int j = 0; j < 4; ++j) { out[i][j + 12] = in[j][4 * i + 3]; }
        }

        kernel.packet[0] = pload<Packet16h>(out[0]);
        kernel.packet[1] = pload<Packet16h>(out[1]);
        kernel.packet[2] = pload<Packet16h>(out[2]);
        kernel.packet[3] = pload<Packet16h>(out[3]);
    }

    template <> struct is_arithmetic<Packet16bf>
    {
        enum
        {
            value = true
        };
    };

    template <> struct packet_traits<bfloat16> : default_packet_traits
    {
        typedef Packet16bf type;
        typedef Packet8bf half;
        enum
        {
            Vectorizable = 1,
            AlignedOnScalar = 1,
            size = 16,
            HasHalfPacket = 1,
            HasBlend = 0,
            HasInsert = 1,
            HasSin = EIGEN_FAST_MATH,
            HasCos = EIGEN_FAST_MATH,
#if EIGEN_GNUC_AT_LEAST(5, 3) || (!EIGEN_COMP_GNUC_STRICT)
#ifdef EIGEN_VECTORIZE_AVX512DQ
            HasLog = 1,  // Currently fails test with bad accuracy.
            HasLog1p = 1,
            HasExpm1 = 1,
            HasNdtri = 1,
            HasBessel = 1,
#endif
            HasExp = 1,
            HasSqrt = EIGEN_FAST_MATH,
            HasRsqrt = EIGEN_FAST_MATH,
            HasTanh = EIGEN_FAST_MATH,
            HasErf = EIGEN_FAST_MATH,
#endif
            HasCmp = 1,
            HasDiv = 1
        };
    };

    template <> struct unpacket_traits<Packet16bf>
    {
        typedef bfloat16 type;
        enum
        {
            size = 16,
            alignment = Aligned32,
            vectorizable = true,
            masked_load_available = false,
            masked_store_available = false
        };
        typedef Packet8bf half;
    };

    template <> EIGEN_STRONG_INLINE Packet16bf pset1<Packet16bf>(const bfloat16& from) { return _mm256_set1_epi16(from.value); }

    template <> EIGEN_STRONG_INLINE bfloat16 pfirst<Packet16bf>(const Packet16bf& from)
    {
        bfloat16 t;
        t.value = static_cast<unsigned short>(_mm256_extract_epi16(from, 0));
        return t;
    }

    template <> EIGEN_STRONG_INLINE Packet16bf pload<Packet16bf>(const bfloat16* from) { return _mm256_load_si256(reinterpret_cast<const __m256i*>(from)); }

    template <> EIGEN_STRONG_INLINE Packet16bf ploadu<Packet16bf>(const bfloat16* from) { return _mm256_loadu_si256(reinterpret_cast<const __m256i*>(from)); }

    template <> EIGEN_STRONG_INLINE void pstore<bfloat16>(bfloat16* to, const Packet16bf& from) { _mm256_store_si256(reinterpret_cast<__m256i*>(to), from); }

    template <> EIGEN_STRONG_INLINE void pstoreu<bfloat16>(bfloat16* to, const Packet16bf& from) { _mm256_storeu_si256(reinterpret_cast<__m256i*>(to), from); }

    template <> EIGEN_STRONG_INLINE Packet16bf ploaddup<Packet16bf>(const bfloat16* from)
    {
        Packet16bf r;
        unsigned short a = from[0].value;
        unsigned short b = from[1].value;
        unsigned short c = from[2].value;
        unsigned short d = from[3].value;
        unsigned short e = from[4].value;
        unsigned short f = from[5].value;
        unsigned short g = from[6].value;
        unsigned short h = from[7].value;
        return _mm256_set_epi16(h, h, g, g, f, f, e, e, d, d, c, c, b, b, a, a);
    }

    template <> EIGEN_STRONG_INLINE Packet16bf ploadquad(const bfloat16* from)
    {
        Packet16bf r;
        unsigned short a = from[0].value;
        unsigned short b = from[1].value;
        unsigned short c = from[2].value;
        unsigned short d = from[3].value;
        return _mm256_set_epi16(d, d, d, d, c, c, c, c, b, b, b, b, a, a, a, a);
    }

    EIGEN_STRONG_INLINE Packet16f Bf16ToF32(const Packet16bf& a) { return _mm512_castsi512_ps(_mm512_slli_epi32(_mm512_cvtepu16_epi32(a), 16)); }

    // Convert float to bfloat16 according to round-to-nearest-even/denormals algorithm.
    EIGEN_STRONG_INLINE Packet16bf F32ToBf16(const Packet16f& a)
    {
        Packet16bf r;

#if defined(EIGEN_VECTORIZE_AVX512BF16) && EIGEN_GNUC_AT_LEAST(10, 1)
        // Since GCC 10.1 supports avx512bf16 and C style explicit cast
        // (C++ static_cast is not supported yet), do converion via intrinsic
        // and register path for performance.
        r = (__m256i)(_mm512_cvtneps_pbh(a));

#else
        __m512i t;
        __m512i input = _mm512_castps_si512(a);
        __m512i nan = _mm512_set1_epi32(0x7fc0);

        // uint32_t lsb = (input >> 16) & 1;
        t = _mm512_and_si512(_mm512_srli_epi32(input, 16), _mm512_set1_epi32(1));
        // uint32_t rounding_bias = 0x7fff + lsb;
        t = _mm512_add_epi32(t, _mm512_set1_epi32(0x7fff));
        // input += rounding_bias;
        t = _mm512_add_epi32(t, input);
        // input = input >> 16;
        t = _mm512_srli_epi32(t, 16);

        // Check NaN before converting back to bf16
        __mmask16 mask = _mm512_cmp_ps_mask(a, a, _CMP_ORD_Q);

        t = _mm512_mask_blend_epi32(mask, nan, t);
        // output.value = static_cast<uint16_t>(input);
        r = _mm512_cvtepi32_epi16(t);
#endif  // EIGEN_VECTORIZE_AVX512BF16

        return r;
    }

    template <> EIGEN_STRONG_INLINE Packet16bf ptrue(const Packet16bf& a) { return ptrue<Packet8i>(a); }

    template <> EIGEN_STRONG_INLINE Packet16bf por(const Packet16bf& a, const Packet16bf& b) { return por<Packet8i>(a, b); }

    template <> EIGEN_STRONG_INLINE Packet16bf pxor(const Packet16bf& a, const Packet16bf& b) { return pxor<Packet8i>(a, b); }

    template <> EIGEN_STRONG_INLINE Packet16bf pand(const Packet16bf& a, const Packet16bf& b) { return pand<Packet8i>(a, b); }

    template <> EIGEN_STRONG_INLINE Packet16bf pandnot(const Packet16bf& a, const Packet16bf& b) { return pandnot<Packet8i>(a, b); }

    template <> EIGEN_STRONG_INLINE Packet16bf pselect(const Packet16bf& mask, const Packet16bf& a, const Packet16bf& b)
    {
        // Input mask is expected to be all 0/1, handle it with 8-bit
        // intrinsic for performance.
        return _mm256_blendv_epi8(b, a, mask);
    }

    template <> EIGEN_STRONG_INLINE Packet16bf pround<Packet16bf>(const Packet16bf& a) { return F32ToBf16(pround<Packet16f>(Bf16ToF32(a))); }

    template <> EIGEN_STRONG_INLINE Packet16bf print<Packet16bf>(const Packet16bf& a) { return F32ToBf16(print<Packet16f>(Bf16ToF32(a))); }

    template <> EIGEN_STRONG_INLINE Packet16bf pceil<Packet16bf>(const Packet16bf& a) { return F32ToBf16(pceil<Packet16f>(Bf16ToF32(a))); }

    template <> EIGEN_STRONG_INLINE Packet16bf pfloor<Packet16bf>(const Packet16bf& a) { return F32ToBf16(pfloor<Packet16f>(Bf16ToF32(a))); }

    template <> EIGEN_STRONG_INLINE Packet16bf pcmp_eq(const Packet16bf& a, const Packet16bf& b) { return Pack32To16(pcmp_eq(Bf16ToF32(a), Bf16ToF32(b))); }

    template <> EIGEN_STRONG_INLINE Packet16bf pcmp_le(const Packet16bf& a, const Packet16bf& b) { return Pack32To16(pcmp_le(Bf16ToF32(a), Bf16ToF32(b))); }

    template <> EIGEN_STRONG_INLINE Packet16bf pcmp_lt(const Packet16bf& a, const Packet16bf& b) { return Pack32To16(pcmp_lt(Bf16ToF32(a), Bf16ToF32(b))); }

    template <> EIGEN_STRONG_INLINE Packet16bf pcmp_lt_or_nan(const Packet16bf& a, const Packet16bf& b)
    {
        return Pack32To16(pcmp_lt_or_nan(Bf16ToF32(a), Bf16ToF32(b)));
    }

    template <> EIGEN_STRONG_INLINE Packet16bf pnegate(const Packet16bf& a)
    {
        Packet16bf sign_mask = _mm256_set1_epi16(static_cast<unsigned short>(0x8000));
        return _mm256_xor_si256(a, sign_mask);
    }

    template <> EIGEN_STRONG_INLINE Packet16bf pconj(const Packet16bf& a) { return a; }

    template <> EIGEN_STRONG_INLINE Packet16bf pabs(const Packet16bf& a)
    {
        const __m256i sign_mask = _mm256_set1_epi16(static_cast<numext::uint16_t>(0x8000));
        return _mm256_andnot_si256(sign_mask, a);
    }

    template <> EIGEN_STRONG_INLINE Packet16bf padd<Packet16bf>(const Packet16bf& a, const Packet16bf& b)
    {
        return F32ToBf16(padd<Packet16f>(Bf16ToF32(a), Bf16ToF32(b)));
    }

    template <> EIGEN_STRONG_INLINE Packet16bf psub<Packet16bf>(const Packet16bf& a, const Packet16bf& b)
    {
        return F32ToBf16(psub<Packet16f>(Bf16ToF32(a), Bf16ToF32(b)));
    }

    template <> EIGEN_STRONG_INLINE Packet16bf pmul<Packet16bf>(const Packet16bf& a, const Packet16bf& b)
    {
        return F32ToBf16(pmul<Packet16f>(Bf16ToF32(a), Bf16ToF32(b)));
    }

    template <> EIGEN_STRONG_INLINE Packet16bf pdiv<Packet16bf>(const Packet16bf& a, const Packet16bf& b)
    {
        return F32ToBf16(pdiv<Packet16f>(Bf16ToF32(a), Bf16ToF32(b)));
    }

    template <> EIGEN_STRONG_INLINE Packet16bf pmin<Packet16bf>(const Packet16bf& a, const Packet16bf& b)
    {
        return F32ToBf16(pmin<Packet16f>(Bf16ToF32(a), Bf16ToF32(b)));
    }

    template <> EIGEN_STRONG_INLINE Packet16bf pmax<Packet16bf>(const Packet16bf& a, const Packet16bf& b)
    {
        return F32ToBf16(pmax<Packet16f>(Bf16ToF32(a), Bf16ToF32(b)));
    }

    template <> EIGEN_STRONG_INLINE Packet16bf plset<Packet16bf>(const bfloat16& a) { return F32ToBf16(plset<Packet16f>(static_cast<float>(a))); }

    template <> EIGEN_STRONG_INLINE Packet8bf predux_half_dowto4<Packet16bf>(const Packet16bf& a)
    {
        Packet8bf lane0 = _mm256_extractf128_si256(a, 0);
        Packet8bf lane1 = _mm256_extractf128_si256(a, 1);
        return padd<Packet8bf>(lane0, lane1);
    }

    template <> EIGEN_STRONG_INLINE bfloat16 predux<Packet16bf>(const Packet16bf& p) { return static_cast<bfloat16>(predux<Packet16f>(Bf16ToF32(p))); }

    template <> EIGEN_STRONG_INLINE bfloat16 predux_mul<Packet16bf>(const Packet16bf& from)
    {
        return static_cast<bfloat16>(predux_mul<Packet16f>(Bf16ToF32(from)));
    }

    template <> EIGEN_STRONG_INLINE bfloat16 predux_min<Packet16bf>(const Packet16bf& from)
    {
        return static_cast<bfloat16>(predux_min<Packet16f>(Bf16ToF32(from)));
    }

    template <> EIGEN_STRONG_INLINE bfloat16 predux_max<Packet16bf>(const Packet16bf& from)
    {
        return static_cast<bfloat16>(predux_max<Packet16f>(Bf16ToF32(from)));
    }

    template <> EIGEN_STRONG_INLINE Packet16bf preverse(const Packet16bf& a)
    {
        __m256i m = _mm256_setr_epi8(14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1);

        Packet16bf res;
        // Swap hi and lo first because shuffle is in 128-bit lanes.
        res = _mm256_permute2x128_si256(a, a, 1);
        // Shuffle 8-bit values in src within 2*128-bit lanes.
        return _mm256_shuffle_epi8(res, m);
    }

    template <> EIGEN_STRONG_INLINE Packet16bf pgather<bfloat16, Packet16bf>(const bfloat16* from, Index stride)
    {
        return _mm256_set_epi16(from[15 * stride].value,
                                from[14 * stride].value,
                                from[13 * stride].value,
                                from[12 * stride].value,
                                from[11 * stride].value,
                                from[10 * stride].value,
                                from[9 * stride].value,
                                from[8 * stride].value,
                                from[7 * stride].value,
                                from[6 * stride].value,
                                from[5 * stride].value,
                                from[4 * stride].value,
                                from[3 * stride].value,
                                from[2 * stride].value,
                                from[1 * stride].value,
                                from[0 * stride].value);
    }

    template <> EIGEN_STRONG_INLINE void pscatter<bfloat16, Packet16bf>(bfloat16* to, const Packet16bf& from, Index stride)
    {
        EIGEN_ALIGN64 bfloat16 aux[16];
        pstore(aux, from);
        to[stride * 0] = aux[0];
        to[stride * 1] = aux[1];
        to[stride * 2] = aux[2];
        to[stride * 3] = aux[3];
        to[stride * 4] = aux[4];
        to[stride * 5] = aux[5];
        to[stride * 6] = aux[6];
        to[stride * 7] = aux[7];
        to[stride * 8] = aux[8];
        to[stride * 9] = aux[9];
        to[stride * 10] = aux[10];
        to[stride * 11] = aux[11];
        to[stride * 12] = aux[12];
        to[stride * 13] = aux[13];
        to[stride * 14] = aux[14];
        to[stride * 15] = aux[15];
    }

    EIGEN_STRONG_INLINE void ptranspose(PacketBlock<Packet16bf, 16>& kernel)
    {
        __m256i a = kernel.packet[0];
        __m256i b = kernel.packet[1];
        __m256i c = kernel.packet[2];
        __m256i d = kernel.packet[3];
        __m256i e = kernel.packet[4];
        __m256i f = kernel.packet[5];
        __m256i g = kernel.packet[6];
        __m256i h = kernel.packet[7];
        __m256i i = kernel.packet[8];
        __m256i j = kernel.packet[9];
        __m256i k = kernel.packet[10];
        __m256i l = kernel.packet[11];
        __m256i m = kernel.packet[12];
        __m256i n = kernel.packet[13];
        __m256i o = kernel.packet[14];
        __m256i p = kernel.packet[15];

        __m256i ab_07 = _mm256_unpacklo_epi16(a, b);
        __m256i cd_07 = _mm256_unpacklo_epi16(c, d);
        __m256i ef_07 = _mm256_unpacklo_epi16(e, f);
        __m256i gh_07 = _mm256_unpacklo_epi16(g, h);
        __m256i ij_07 = _mm256_unpacklo_epi16(i, j);
        __m256i kl_07 = _mm256_unpacklo_epi16(k, l);
        __m256i mn_07 = _mm256_unpacklo_epi16(m, n);
        __m256i op_07 = _mm256_unpacklo_epi16(o, p);

        __m256i ab_8f = _mm256_unpackhi_epi16(a, b);
        __m256i cd_8f = _mm256_unpackhi_epi16(c, d);
        __m256i ef_8f = _mm256_unpackhi_epi16(e, f);
        __m256i gh_8f = _mm256_unpackhi_epi16(g, h);
        __m256i ij_8f = _mm256_unpackhi_epi16(i, j);
        __m256i kl_8f = _mm256_unpackhi_epi16(k, l);
        __m256i mn_8f = _mm256_unpackhi_epi16(m, n);
        __m256i op_8f = _mm256_unpackhi_epi16(o, p);

        __m256i abcd_03 = _mm256_unpacklo_epi32(ab_07, cd_07);
        __m256i abcd_47 = _mm256_unpackhi_epi32(ab_07, cd_07);
        __m256i efgh_03 = _mm256_unpacklo_epi32(ef_07, gh_07);
        __m256i efgh_47 = _mm256_unpackhi_epi32(ef_07, gh_07);
        __m256i ijkl_03 = _mm256_unpacklo_epi32(ij_07, kl_07);
        __m256i ijkl_47 = _mm256_unpackhi_epi32(ij_07, kl_07);
        __m256i mnop_03 = _mm256_unpacklo_epi32(mn_07, op_07);
        __m256i mnop_47 = _mm256_unpackhi_epi32(mn_07, op_07);

        __m256i abcd_8b = _mm256_unpacklo_epi32(ab_8f, cd_8f);
        __m256i abcd_cf = _mm256_unpackhi_epi32(ab_8f, cd_8f);
        __m256i efgh_8b = _mm256_unpacklo_epi32(ef_8f, gh_8f);
        __m256i efgh_cf = _mm256_unpackhi_epi32(ef_8f, gh_8f);
        __m256i ijkl_8b = _mm256_unpacklo_epi32(ij_8f, kl_8f);
        __m256i ijkl_cf = _mm256_unpackhi_epi32(ij_8f, kl_8f);
        __m256i mnop_8b = _mm256_unpacklo_epi32(mn_8f, op_8f);
        __m256i mnop_cf = _mm256_unpackhi_epi32(mn_8f, op_8f);

        __m256i abcdefgh_01 = _mm256_unpacklo_epi64(abcd_03, efgh_03);
        __m256i abcdefgh_23 = _mm256_unpackhi_epi64(abcd_03, efgh_03);
        __m256i ijklmnop_01 = _mm256_unpacklo_epi64(ijkl_03, mnop_03);
        __m256i ijklmnop_23 = _mm256_unpackhi_epi64(ijkl_03, mnop_03);
        __m256i abcdefgh_45 = _mm256_unpacklo_epi64(abcd_47, efgh_47);
        __m256i abcdefgh_67 = _mm256_unpackhi_epi64(abcd_47, efgh_47);
        __m256i ijklmnop_45 = _mm256_unpacklo_epi64(ijkl_47, mnop_47);
        __m256i ijklmnop_67 = _mm256_unpackhi_epi64(ijkl_47, mnop_47);
        __m256i abcdefgh_89 = _mm256_unpacklo_epi64(abcd_8b, efgh_8b);
        __m256i abcdefgh_ab = _mm256_unpackhi_epi64(abcd_8b, efgh_8b);
        __m256i ijklmnop_89 = _mm256_unpacklo_epi64(ijkl_8b, mnop_8b);
        __m256i ijklmnop_ab = _mm256_unpackhi_epi64(ijkl_8b, mnop_8b);
        __m256i abcdefgh_cd = _mm256_unpacklo_epi64(abcd_cf, efgh_cf);
        __m256i abcdefgh_ef = _mm256_unpackhi_epi64(abcd_cf, efgh_cf);
        __m256i ijklmnop_cd = _mm256_unpacklo_epi64(ijkl_cf, mnop_cf);
        __m256i ijklmnop_ef = _mm256_unpackhi_epi64(ijkl_cf, mnop_cf);

        // NOTE: no unpacklo/hi instr in this case, so using permute instr.
        kernel.packet[0] = _mm256_permute2x128_si256(abcdefgh_01, ijklmnop_01, 0x20);
        kernel.packet[1] = _mm256_permute2x128_si256(abcdefgh_23, ijklmnop_23, 0x20);
        kernel.packet[2] = _mm256_permute2x128_si256(abcdefgh_45, ijklmnop_45, 0x20);
        kernel.packet[3] = _mm256_permute2x128_si256(abcdefgh_67, ijklmnop_67, 0x20);
        kernel.packet[4] = _mm256_permute2x128_si256(abcdefgh_89, ijklmnop_89, 0x20);
        kernel.packet[5] = _mm256_permute2x128_si256(abcdefgh_ab, ijklmnop_ab, 0x20);
        kernel.packet[6] = _mm256_permute2x128_si256(abcdefgh_cd, ijklmnop_cd, 0x20);
        kernel.packet[7] = _mm256_permute2x128_si256(abcdefgh_ef, ijklmnop_ef, 0x20);
        kernel.packet[8] = _mm256_permute2x128_si256(abcdefgh_01, ijklmnop_01, 0x31);
        kernel.packet[9] = _mm256_permute2x128_si256(abcdefgh_23, ijklmnop_23, 0x31);
        kernel.packet[10] = _mm256_permute2x128_si256(abcdefgh_45, ijklmnop_45, 0x31);
        kernel.packet[11] = _mm256_permute2x128_si256(abcdefgh_67, ijklmnop_67, 0x31);
        kernel.packet[12] = _mm256_permute2x128_si256(abcdefgh_89, ijklmnop_89, 0x31);
        kernel.packet[13] = _mm256_permute2x128_si256(abcdefgh_ab, ijklmnop_ab, 0x31);
        kernel.packet[14] = _mm256_permute2x128_si256(abcdefgh_cd, ijklmnop_cd, 0x31);
        kernel.packet[15] = _mm256_permute2x128_si256(abcdefgh_ef, ijklmnop_ef, 0x31);
    }

    EIGEN_STRONG_INLINE void ptranspose(PacketBlock<Packet16bf, 4>& kernel)
    {
        __m256i a = kernel.packet[0];
        __m256i b = kernel.packet[1];
        __m256i c = kernel.packet[2];
        __m256i d = kernel.packet[3];

        __m256i ab_07 = _mm256_unpacklo_epi16(a, b);
        __m256i cd_07 = _mm256_unpacklo_epi16(c, d);
        __m256i ab_8f = _mm256_unpackhi_epi16(a, b);
        __m256i cd_8f = _mm256_unpackhi_epi16(c, d);

        __m256i abcd_03 = _mm256_unpacklo_epi32(ab_07, cd_07);
        __m256i abcd_47 = _mm256_unpackhi_epi32(ab_07, cd_07);
        __m256i abcd_8b = _mm256_unpacklo_epi32(ab_8f, cd_8f);
        __m256i abcd_cf = _mm256_unpackhi_epi32(ab_8f, cd_8f);

        // NOTE: no unpacklo/hi instr in this case, so using permute instr.
        kernel.packet[0] = _mm256_permute2x128_si256(abcd_03, abcd_47, 0x20);
        kernel.packet[1] = _mm256_permute2x128_si256(abcd_8b, abcd_cf, 0x20);
        kernel.packet[2] = _mm256_permute2x128_si256(abcd_03, abcd_47, 0x31);
        kernel.packet[3] = _mm256_permute2x128_si256(abcd_8b, abcd_cf, 0x31);
    }

}  // end namespace internal

}  // end namespace Eigen

#endif  // EIGEN_PACKET_MATH_AVX512_H
